ÉTALE CHERN CLASSES AT THE PRIME 2

CHARLES WEIBEL

Department of Mathematics Rutgers University New Brunswick, NJ 08903

Abstract. Let A be a commutative ring and q a power of 2. We investigate the étale Chern classes

c_{ik} : K_{n}(A; ℤ/q) ⟶ H_{et}^{k}(A; μ_{q}^{⊗i}),

which are defined whenever i ≥ 1 and n + k = 2i. These are group homomorphisms except when n = 2 and q is even. The usual product formula for c_{ik}({a, b}) remains valid, except when q = 2 and i ≥ 3, when there is a correction term.

Introduction.

Let A be a commutative ring. For any integer q, let K_{∗}(A; ℤ/q) denote the mod q K-theory groups of A, and let μ_{q} denote the étale sheaf of q^{th} roots of unity on Spec(A). In his 1979 paper [S], Soulé constructed étale Chern classes

c_{ik} : K_{n}(A; ℤ/q) ⟶ H_{et}^{k}(A; μ_{q}^{⊗i}), n + k = 2i, i ≥ 1

for any q, and proved some interesting things about the c_{ik} when q is odd. Recently, there has been an interest in these classes when q = 2^{ν}. The purpose of this paper is to describe what happens, and to provide correction terms for Soulé’s results in this case.

Our most dramatic results may be illustrated with the mod 2 K-theory of the real numbers ℝ. It is well-known [Wp, p.396] [AT,p.79] that K_{2}(ℝ) ⊗ ℤ/2 ≅ ℤ/2 is a subgroup of K_{2}(ℝ; ℤ/2) ≅ ℤ/4. Let β ∈ K_{2}(ℝ; ℤ/2) be a generator, so that 2β is the image of the nonzero element {−1, −1} of K_{2}(ℝ) ⊗ ℤ/2 ≅ ℤ/2. (0.1) The target of the Chern class c_{22} is the Brauer group H_{et}^{2}(ℝ; μ_{2}^{⊗i}) = Br(ℝ) ≅

ℤ/2. One would think that the map c_{22} : ℤ/4 → ℤ/2 would send 2β = {−1, −1} to zero. Yet we know (say from [Sh]) that c_{22} is the Galois symbol on K_{2}; since c_{22}({−1, −1}) is represented by the class of the quaternions ℍ in the Brauer group of ℝ, it is nonzero! The resolution of this paradox is that c_{22} is not a group homomorphism!

(0.2) Suslin asserts in [Sus2, p.13] that Soulé’s product formula remains valid for q = 2^{ν}, provided that we consider products of the type

K_{m}(A) × K_{n}(A; ℤ/q) ⟶ K_{m+n}(A; ℤ/q).

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Partially supported by NSF grants

On the other hand, we will see that

\begin{aligned}c_{33}(\{−1, β\}) & = \{−1, −1, −1\} ≠ 0 \\ c_{44}(\{−1, −1, β\}) & = \{−1, −1, −1, −1\} ≠ 0 \end{aligned}  \begin{matrix}in H_{et}^{3}(ℝ; ℤ/2) ≅ ℤ/2; \\ in H_{et}^{4}(ℝ; ℤ/2) ≅ ℤ/2. \end{matrix}

In both cases, the product formula gives zero. The resolution of this paradox is that the product formula requires a correction term when i ≥ 3, m = 2 and coefficients are taken in ℤ/2. Fortunately, the only applications of Suslin’s assertion to date (to our knowledge) have been to use the correct formula for the Chern class c_{21} on K_{3}(F; ℤ/q): if ζ = ∂β is a q^{th} root of unity and f ∈ F^{∗},

c_{21}({f, β}) = −{f, ζ} in H_{et}^{1}(F; μ_{q}^{⊗2}).

See [Sus2, 3.2], [MS, 7.1, 7.2], [L, 4.5, 5.1] as well as [K] and [PW, 4.5]. (0.3) Consider the noncommutative diagram

K_{3}(ℝ(t); ℤ/q) ⟶^{∂} K_{2}(ℝ; ℤ/2)
c_{33}↓  ^{c_{22}}↓
H^{3}(ℝ(t); μ_{q}^{⊗3}) ⟶^{∂} H^{2}(ℝ; μ_{q}^{⊗2})

Here the maps ∂ come from localization sequences for the discrete valuation ring ℝ[t]_{(t)}. If q is odd, Soulé proved that ∂c_{33} = −2c_{22}∂. This formula does not hold for q = 2, because in fact:

∂c_{33}({t, β}) = {−1, −1} = c_{22}(∂{t, β}) ≠ 0.

Here the difficulty is that Soulé’s proof uses the product formula for c_{33}. We will see that if 4|q then Soulé’s formula for localization sequences continues to hold:

∂c_{ik} = (1 − i)c_{i−1,k−1}∂.

(0.4) Let A = ℝ[x, y]/(x^{2} +y^{2} = 1) be the coordinate ring of the circle. The natural continuation of the diagram of (0.3), described in 4.3.1, would be the diagram

K_{2}(ℝ; ℤ/2) ⟶^{τ} K_{2}(A; ℤ/2)
↓^{c_{22}}  ↓^{c_{34}}
H^{2}(ℝ; ℤ/2) ⟶ H^{4}(A; ℤ/2).

Here τ is the transfer associated to a point on the circle. By now, the reader should not be surprised to know that something is amiss. In fact c_{34}τ(β) ≠ 0 but c_{22}(β) = 0. What is surprising is not that there is a correction term for the product formula for c_{34}({a, β}) with a ∈ K_{0}(A) – there is – but that the product formula given in [S] is actually off by a factor of ±i_{1}.

(0.5) Let F be a number field and let w_{i} = w_{i}(F) denote the order of the finite cyclic group H_{et}^{0}(F; ℚ/ℤ(i)). Harris and Segal found a natural cyclic direct summand E^{i} of K_{2i−1}(F) whose order was either w_{i}, 2w_{i} or w_{i}/2. We define the (real and complex) e-invariants and show that they detect all of E^{i}; this eliminates some of the uncertainty in the order of the summand E^{i}.

When F is totally imaginary and i ≠ 0 (mod 4), we show that the order of E^{i} is always w_{i}. If 4|i the order is either w_{i} or \frac{1}{2}w_{i}. (See 6.5 and 6.6.)

When F = ℚ and i = 2 + 4k, the uncertainty in E^{i} ⊆ K_{8k+3}(ℚ) was clarified in [Br]: the order is 2w_{i}. We clarify things even further: if i = 4k the order is w_{i}, because E^{i} = Im(J)_{8k−1}. We expect, but cannot quite prove, that the same is true for any number field F having a real embedding (see 6.8).

(0.6) Let A be the ring of ℤ[\frac{1}{2}]-integers in a totally real number field F. Lichtenbaum’s conjectures for the prime ℓ = 2 involve the 2-Sylow subgroups H_{et}^{k}(A; W^{(i)}) of the groups H_{et}^{k}(A; ℚ/ℤ(i)). We observe that whenever k + i is odd (k ≥ 3 and i ≥ 1) these groups are (ℤ/2)^{r_{1}} ≠ 0. This quantifies how periodically false the conjecture (L1) of [Licht] was at ℓ = 2. On the other hand we show in 7.3 that (L1) holds for totally imaginary number fields F, in the sense that for all i ≥ 2 we have

H_{et}^{2}(A; W^{(i)}) = H_{et}^{2}(F; W^{(i)}) = 0.

The organization of this paper parallels the organization of Soulé’s paper [S]. In §1 we study the mod q Hurewicz maps h_{n} : π_{n}(X; ℤ/q) → H_{n}(X; ℤ/q). We show that h_{2} is not always a homomorphism when q is even. We also discuss the behavior of the Hurewicz map with respect to products. Using this, §2 introduces Soulé’s chern classes on K_{n}(A; ℤ/q). These are homomorphisms except when n = 2 and q is even, when there is a correction term. This resolves paradox (0.1) above.

In §3 we discuss (0.2), providing the correction term for Soulé’s product formulas (3.1) and (3.10). This allows us to extend Soulé’s results on localization and transfer in §4, resolving (0.3) and (0.4). Since it fits naturally here, we give a short proof, due to Soulé, that if A is the ring of integers A in a global field F then K_{n}(A) = K_{n}(F) for all odd n. This extension of [S, III.3] is well-known to the experts, and written down in [Sherm] and [S2].

In §5 we study the image of the Chern classes, using [MS], [MS2] and [L] to update the results of [S, IV]. Of course at ℓ = 2 the image of the Chern class c_{ik} tends to have index (i − 1)!, so it is less interesting than in the cases studied by Soulé. When i = 0 we can effectively divide the Chern class c_{i0} by (i−1)! to obtain the e-invariant, studied in §6. The relation with the Harris-Segal summands E^{i} of (0.5) is analyzed.

In §7 we address point (0.6). With the exception of 7.3, which is the 2-primary analogue of Soulé’s proof of (L1) for ℓ = 2, most of this material should be unsurprising to the experts.

The author would like to thank the numerous people who have contributed their assistance to this enterprise: M. Artin, R. Bumby, W. Dwyer, L.Evans, S. Geller, J.F. Jardine, B. Kahn, M. Levine, S. Lichtenbaum, M. Mahowald, N. Pitt, W. Raskind, C. Sherman, R. Thomason and A. Wiles.

1. The Hurewicz “morphism” at the prime 2.

Fix an integer q and let P^{n} denote the n-dimensional mod q Moore space. Recall that the topological space P^{n} is defined only for n ≥ 2, and may be obtained from the sphere S^{n−1} by attaching an n-cell by a map of degree q. The mod q homotopy “groups” of a pointed space X are defined by

π_{n}(X; ℤ/q) = [P^{n}, X], n ≥ 2.

If n ≥ 3 then P^{n} = ΣP^{n−1} is a co-H-space and therefore π_{n}(X; ℤ/q) is a group. If X is an H-space, then π_{2}(X; ℤ/q) is also a group. In either case, the quotient map P^{n} → S^{n} yields a homomorphism π_{n}(X) → π_{n}(X; ℤ/q).

The Bockstein map ∂ : π_{n}(X; ℤ/q) → π_{n−1}(X) is obtained by composing with the inclusion S^{n−1} ⊂ P^{n}. Again supposing either that n ≥ 3 or that n = 2 and X is an H-space, ∂ is a group homomorphism and its image is the q-torsion subgroup of π_{n−1}(X). This all fits together into a universal coefficient theorem [N, 1.4]:

0 ⟶ π_{n}(X) ⊗ ℤ/q ⟶ π_{n}(X; ℤ/q) ⟶^{∂} Tor(π_{n−1}(X), ℤ/q) ⟶ 0.

The mod q Hurewicz map is the set map

h_{n} : π_{n}(X; ℤ/q) ⟶ H_{n}(X; ℤ/q)

which sends α ∈ [P^{n}, X] to α_{∗}(ε_{n}), the image of the canonical generator ε_{n} ∈ H_{n}(P^{n}; ℤ/q). By definition, ε_{n} corresponds to 1 ∈ ℤ/q under the natural isomorphism H_{n}(P^{n}; ℤ/q) ≅ H_{n}(S^{n}; ℤ/q) ≅ ℤ/q induced by the quotient map P^{n} → S^{n}. If n ≥ 3 it is easy to see that h_{n} is a group homomorphism [N, 3.3].

Now suppose that n = 2 and that X is an H-space. If q is odd, h_{2} is also a homomorphism (see 1.2 below). When q is even, h_{2} need not be a group homomorphism, because the image of h_{2} may not be primitive when q is even. To see this, write H_{∗} for H_{∗}( ; ℤ/q), so that the diagonal map Δ_{∗} goes from H_{∗}(X) to H_{∗}(X × X) ≅ H_{∗}(X) ⊗ H_{∗}(X). An element h ∈ H_{∗} is called primitive if Δ_{∗}(h) = 1 ⊗ h + h ⊗ 1. If n ≥ 3 the image of h_{n} consists of primitive elements [N, 3.4].

Let h_{1} : π_{1}(X) → H_{1}(X) denote the integral Hurewicz map, reduced modulo q. We shall need to evaluate h_{1}(∂α), where α ∈ π_{2}(X; ℤ/q). If ε̃_{1} is the generator of H_{1}(S^{1}) and ι : S^{1} ⊂ P^{2} is the standard inclusion, then ε_{1} = ι_{∗}(ε̃_{1}) is a canonical generator of the group H_{1}(P^{2}) ≅ ℤ/q. We have:

h_{1}(∂α) = h_{1}(αι) = (αι)_{∗}(ε̃_{1}) = α_{∗}(ε_{1}).

Proposition 1.1. Suppose that α ∈ π_{2}(X; ℤ/q). When q is odd, the element h_{2}(α) ∈ H_{2}(X; ℤ/q) is primitive. When q is even then

Δ_{∗}(h_{2}α) = (h_{2}α) ⊗ 1 + 1 ⊗ (h_{2}α) + (\frac{q}{2})h_{1}(∂α) ⊗ h_{1}(∂α).

In particular, the element ε_{2} = h_{2}(id) of H_{2}(P^{2}) is not primitive for \underline{any} even q, because H_{2}(P^{2} ⊗ P^{2}) ≅ (ℤ/q)^{3} on basis {ε_{2} ⊗ 1, 1 ⊗ ε_{2}, ε_{1} ⊗ ε_{1}}, and

Δ_{∗}(ε_{2}) = ε_{2} ⊗ 1 + 1 ⊗ ε_{2} + (\frac{q}{2})ε_{1} ⊗ ε_{1}.

Proof. Since Δ_{∗}α_{∗} = (α_{∗} ⊗ α_{∗})Δ_{∗}, it suffices to assume that X = P^{2} and compute Δ_{∗}(ε_{2}). Set G = ℤ/q with generator t. The canonical embedding of P^{2} in the classifying space BG is an injection on H_{∗}, so it suffices to compute the diagonal map from H_{∗}(G) = H_{∗}(BG) to H_{∗}(G × G). Let C_{∗}(G) ⟶^{ε} ℤ be the G-projective resolution of ℤ:

⋯ ℤG ⟶^{1−t} ℤG ⟶^{N} ℤG ⟶^{1−t} ℤG ⟶^{ε} ℤ ⟶ 0

so that C_{∗}(G) ⊗ C_{∗}(G) is a (G × G)-projective resolution of ℤ. To compute Δ_{∗} we need a G-invariant chain map Δ : C_{∗}(G) ⟶ C_{∗}(G) ⊗ C_{∗}(G) lifting the identity on ℤ. If e_{i} is a generator of C_{i}(G), we may find a lifting with Δ(e_{0}) = e_{0} ⊗ e_{0}, Δ(e_{1}) = e_{1} ⊗ e_{0} + te_{0} ⊗ e_{1} and

Δ(e_{2}) = e_{2} ⊗ e_{0} + e_{0} ⊗ e_{2} + ∑_{i<j} t^{i}e_{1} ⊗ t^{j}e_{1}.

(We leave it as a combinational exercise for the reader to check that ∂Δ(e_{1}) = e_{0} ⊗ e_{0} − te_{0} ⊗ te_{0} and ∂Δ(e_{2}) = Δ(Ne_{1}).) Passing to homology yields the calculation:

Δ_{∗}(ε_{2}) = ε_{2} ⊗ 1 + 1 ⊗ ε_{2} + ({q \atop 2})ε_{1} ⊗ ε_{1}.

Finally, observe that if q is odd then ({q \atop 2}) ≡ 0 (mod q), while if q is even then ({q \atop 2}) ≡ q/2 (mod q).

In order to state the next result, write • for the homology product on the ring H_{∗}(X; ℤ/q). If α, β ∈ π_{2}(X; ℤ/q), write {∂α, ∂β} for μ(∂α ∧ ∂β). Note that

h_{2}({∂α, ∂β}) = h_{1}(∂α) • h_{1}(∂β) = α_{∗}(ε_{1}) • β_{∗}(ε_{1})

because the integral Hurewicz homomorphisms commute with products.

Theorem 1.2. Let X be an H-space. If q is odd, the Hurewicz map

h_{2} : π_{2}(X; ℤ/q) ⟶ H_{2}(X; ℤ/q)

is a group homomorphism. If q is even we have

h_{2}(α + β) = h_{2}(α) + h_{2}(β) + \frac{q}{2}h_{2}(\{∂α, ∂β\})

Proof. h_{2}(α + β) is the image of ε_{2} under the composition

H_{2}(P^{2}) ⟶^{Δ_{∗}} H_{2}(P^{2} × P^{2}) ⟶^{(α×β)_{∗}} H_{2}(X × X) ⟶^{μ_{∗}} H_{2}(X).

By the following proposition, we compute, with λ = ({q \atop 2}):

h_{2}(α + β) = μ_{∗}(α × β)_{∗}(ε_{2} ⊗ 1 + 1 ⊗ ε_{2} + λε_{1} ⊗ ε_{1})
= μ_{∗}{α_{∗}(ε_{2}) ⊗ 1 + 1 ⊗ β_{∗}(ε_{2}) + λα_{∗}(ε_{1}) ⊗ β_{∗}(ε_{1})}
= h_{2}(α) + h_{2}(β) + λα_{∗}(ε_{1}) • β_{∗}(ε_{1}).

Example 1.3. Let X be either of the H-spaces BO or ℝℙ^{∞}. It is well known [AT, 1.1] that π_{2}(X; ℤ/2) ≅ ℤ/4 on the standard inclusion ι : P^{2} ⊂ X and that h_{2}(ι) = ε_{2}. Since h_{2}(0) = 0, the theorem gives us

h_{2}(2ι) = 2h_{2}(ι) + ε_{1} • ε_{1} = ε_{1} • ε_{1} = ε_{2}.
h_{2}(−ι) = h_{2}(ι) + h_{2}(2ι) + ε_{1} • 0 = 2ε_{2} = 0.

Example 1.4. The mod q K-theory of an associative ring A is defined to be K_{n}(A; ℤ/q) = π_{n}(X; ℤ/q) for the space X = BGL(A)^{+}. For n ≥ 2, the mod q Hurewicz map goes from

h_{n} : K_{n}(A; ℤ/q) ⟶ H_{n}(X; ℤ/q) ≅ H_{n}(GL(A); ℤ/q).

This is a group homomorphism unless n = 2 and q is even. The failure of the Hurewicz map h_{2} to be a homomorphism underlies the Chern class paradox mentioned in the introduction. Indeed, when q = 2 there is an element β ∈ K_{2}(A; ℤ/2) such that ∂β = [−1] ∈ K_{1}(A); h_{2} fails to be a homomorphism whenever h_{2}(β+β) = h_{2}({−1, −1}) is not equal to 2h_{2}(β) = 0.

When q is even and 4|q, we can do slightly better, using the following elementary lemma.

Lemma 1.4.1. If F is a ℤ[\frac{1}{2}]-algebra and x, y ∈ μ_{q}(F), then 2\{x, y\} = 0 in K_{2}(F).

Proof. It is classical that 2{t, t} = 2{−1, t} = 0 and hence that 2{t^{i}, t^{j}} for every t ∈ F^{∗}. If F is a domain then μ_{q}(F) is cyclic, and the result follows. In general, we may assume that F is finitely generated over ℤ[\frac{1}{2}] and restrict to each component of F to assume F has only one minimal prime ideal. But then μ_{q}(F) = μ_{q}(F_{red}) is again cyclic, and the result follows.

Corollary 1.4.2. (4|q) Let F be a ℤ[\frac{1}{2}]-algebra such that K_{1}(F) = F^{∗}. If 4|q then the Hurewicz map

h_{2} : K_{2}(F; ℤ/q) ⟶ H_{2}(GL(F); ℤ/q)

is a group homomorphism.

To conclude this section, we discuss the behavior of the mod q Hurewicz map under products. Assume we are given a pairing of spaces μ : X ∧ Y → Z. For example, if X = BGL(A)^{+} we can consider Loday’s pairing X ∧ X → X. There is a bilinear pairing π_{∗}(X) ⊗ π_{∗}(Y) → π_{∗}(Z) sending a ⊗ b to {a, b} = μ(a ∧ b). If q is odd or 4|q, there is also a bilinear pairing

μ : π_{∗}(X; ℤ/q) ⊗ π_{∗}(Y; ℤ/q) → π_{∗}(Z; ℤ/q)

(see [Br, 1.6]). In Lemma 2 of [S], Soulé proves that this pairing fits into a commutative diagram (for m, n ≥ 2):

(1.5)  \begin{matrix}π_{m}(X; ℤ/q) × π_{n}(Y; ℤ/q) & ⟶^{μ} & π_{m+n}(Z; ℤ/q) \\ ^{h_{m}×h_{n}}↓ & & ^{h_{m+n}}↓ \\ H_{m}(X; ℤ/q) ⊗ H_{n}(Y; ℤ/q) & ⟶^{μ} & H_{m+n}(Z; ℤ/q). \end{matrix}

There is no ⊗ in the upper left corner of (1.5) because h_{2} need not be a homomorphism when q is even. We shall refer to this property as “bilinearity,” even though this is a slight abuse of terminology.

When q = 2 there is no such product on π_{∗}( ; ℤ/2), essentially because P^{m} ∧ P^{n} no longer decomposes as P^{m+n} ∨ P^{m+n−1}. However, there is still a pairing

π_{m}(X) × π_{n}(Y; ℤ/2) ⟶ π_{m+n}(Z; ℤ/2), (m ≥ 0, n ≥ 2)

arising from the isomorphism S^{m} ∧ P^{n} ≅ P^{m+n}; again (1.5) commutes. More generally, let π_{m} × π_{n} denote the following subset of π_{m}(X; ℤ/2) × π_{n}(Y; ℤ/2):

π_{m} × π_{n} = {(α, β) : {∂α, ∂β} = 0 in π_{m+n−2}(Z)} .

Proposition 1.6. (Mod 2 products) Suppose given a pairing of spaces X ∧Y → Z. Then we can define a partial pairing μ : π_{m} × π_{n} → π_{m+n}(Z; ℤ/2) such that the following diagram commutes:

π_{m} × π_{n}  ⟶^{μ} π_{m+n}(Z; ℤ/2)
h_{m}×h_{n}↓  ^{h_{m+n}}↓
H_{m}(X; ℤ/2) × H_{n}(Y; ℤ/2) ⟶^{μ_{∗}} H_{m+n}(Z; ℤ/2).

This partial pairing is well-defined and bilinear if π_{m+n}(Z) has no 2-torsion, or more generally if the Hopf map π_{m+n−1}(Z) → π_{m+n}(Z) is zero.

Proof. (see [Br, 1.9]) If α ∈ π_{m}(X; ℤ/2) and β ∈ π_{n}(Y; ℤ/2) the element {∂α, ∂β} represents the composition

S^{m+n−2} = S^{m−1} ∧ S^{n−1} ⊂ P^{m} ∧ P^{n} ⟶^{α∧β} X ∧ Y ⟶^{μ} Z.

Set Q = (P^{m} ∧ P^{n})/(S^{m+n−2}) ≃ P^{m+n} ∨ S^{m+n−1}. If {∂α, ∂β} = 0 in π_{m+n−2}(Z) then μ(α ∧ β) factors through a map φ : Q → Z. Any other choice of φ will differ by a map Q → S^{m+n−1} → Z. Since the composition η : P^{m+n} ⊂ Q → S^{m+n−1} is P^{m+n} → S^{m+n} followed by the Hopf map, the restriction φ|P^{m+n} represents an element {α, β} ∈ π_{m+n}(Z; ℤ/2) which is well-defined up to an element of the form γ ∘ η, where γ ∈ π_{m+n−1}(Z). We set μ(α, β) = {α, β}. Note that h_{m+n}({α, β}) ∈ H_{m+n}(Z; ℤ/2) is independent of this choice because h_{m+n}(γ ∘ η) = 0.

To see that the diagram commutes, we first suppose that the pairing is the natural quotient μ′ : P^{m} ∧ P^{n} → Q and α, β are the identity maps. In this case we can take Q → Z to be the identity map, and represent {id_{m}, id_{n}} ∈ π_{m+n}(Q; ℤ/2) by the inclusion P^{m+N} ⊂ Q. Clearly h_{m+n}({id_{m}, id_{n}}) is the natural generator ε_{m+n} = μ′_{∗}(ε_{m} ⊗ ε_{n}) of H_{m+n}(Q; ℤ/2).

For a general pairing μ, suppose given α, β so that {∂α, ∂β} = 0. The commutativity of (1.6) follows from the calculation:

h_{m+n}({α, β}) = φ_{∗}(μ′_{∗}(ε_{m} ⊗ ε_{n})) = μ_{∗}(α ∧ β)_{∗}(ε_{m} ∧ ε_{n}) = μ_{∗}(h_{m}(α) ⊗ h_{n}(β)).

Corollary 1.6.1. (Browder [Br, 1.9]) Suppose that there is no 2-torsion in the even homotopy groups π_{2m}(X) of a space X. Then any pairing X ∧ X → X induces a well-defined product on π_{∗}(X; ℤ/2) such that (1.5) commutes.

Indeed, if m + n is odd then either ∂α or ∂β is zero.

Corollary 1.6.2. (Suslin [Sus1]) Let k be an algebraically closed field containing 1/q, and let β ∈ K_{2} be a “Bott” element, i.e., an element such that ∂β is a primitive q^{th} root of unity. Then there are ring isomorphisms for all q, including q = 2:

K_{∗}(k; ℤ/q) ≅ ℤ/q[β].

Corollary 1.6.3. (Browder [Br, 2.6]) If 𝔽 is a finite field of characteristic ≠ 2 then

K_{∗}(𝔽; ℤ/2) ≅ ℤ/2[η, β]/(η^{2} = 0),

where η ∈ K_{1} is the class of the unit −1 and β ∈ K_{2} is the “Bott” element.

Example 1.6.4. The graded abelian group K_{∗}(ℝ; ℤ/2) is determined in [Sus1]; prominent elements are the class η ∈ K_{1} of the unit −1, the “Bott” element β ∈ K_{2} and the “periodicity” element γ ∈ K_{8}. By (1.6) K_{∗}(ℝ; ℤ/2) contains the commutative ring

ℤ/2[η, γ]/(η^{3} = 0),

The product {β, γ} exists in K_{10}, but is only defined up to a multiple of η^{2}γ = {−1, −1, γ}. The product {β, β} does not exist, and we only have a partial product π_{2} × π_{2} → K_{4}(ℝ; ℤ/2), where π_{2} × π_{2} is the set of all pairs (x, y) of elements in K_{2}(ℝ; ℤ/2) ≅ ℤ/4 except (β, β). The group K_{4}(ℝ; ℤ/2) ≅ ℤ/2 is generated by η^{2}β = {−1, −1, β}, but the product K_{4}(ℝ; ℤ/2) ⊗ K_{4}(ℝ; ℤ/2) → K_{8}(ℝ; ℤ/2) is zero.

2. Chern classes at the prime 2.

Let A be a commutative ring containing 1/ℓ and set q = ℓ^{ν}. Chern classes in K-theory may be constructed using Grothendieck’s theory of Chern classes for representations in [G]. For every representation ρ : G ⟶ GL_{n}(A) of a group G Grothendieck constructs elements c_{i}(ρ) ∈ H_{et}^{2i}(A, G; μ_{q}^{⊗i}). Via the Künneth homomorphism

Φ : H_{et}^{2i}(A, G; μ_{q}^{⊗i}) ⟶ ∐_{k=0}^{2i} Hom (H_{2i−k}(G; ℤ/q), H_{et}^{k}(A; μ_{q}^{⊗i}))

these elements yield homomorphisms c_{ik}(ρ) from H_{2i−k}(G; ℤ/q) to H_{et}^{k}(A; μ_{q}^{⊗i}); see [S, II.1]. Taking G = GL_{N}(A) and ρ = id_{N}, and passing to the limit as N → ∞, we obtain homomorphisms

c_{ik}(id) : H_{n}(GL(A); ℤ/q) ⟶ H_{et}^{k}(A, μ_{q}^{⊗i}), n + k = 2i.

Composing with the Hurewicz map h_{n} of (1.4) yields Soulé’s Chern classes,

(2.1)  c_{ik} : K_{n}(A; ℤ/q) ⟶ H_{et}^{k}(A, μ_{q}^{⊗i}),  n + k = 2i, n ≥ 2.

Proposition 2.1.1. The Chern classes are natural in A. That is, if f : A → B is a ring homomorphism then the following diagram commutes:

K_{n}(A; ℤ/q) ⟶^{c_{ik}} H_{et}^{k}(A; μ_{q}^{⊗i})
↓  ↓
K_{n}(B; ℤ/q) ⟶^{c_{ik}} H_{et}^{k}(B; μ_{q}^{⊗i})

Proof. Set G_{1} = GL_{N}(A) and G_{2} = GL_{N}(B). Given compatible representations

G_{1} ⟶^{ρ_{1}} GL_{n}(A)
φ↓  ^{f}↓
G_{2} ⟶^{ρ_{2}} GL_{n}(B)

we have a commutative diagram:

H_{et}^{2i}(A, G_{1}; μ_{q}^{⊗i}) ⟶^{f^{∗}} H_{et}^{2i}(B, G_{1}; μ_{q}^{⊗i}) ⟵^{φ^{∗}}  H^{2i}(B, G_{2}; μ_{q}^{⊗i})
Φ↓  ^{Φ}↓  ^{Φ}↓
Hom(H_{n}(G_{1}), H^{k}(A)) ⟶^{f^{∗}} Hom(H_{n}(G_{1}), H^{k}(B)) ⟵^{φ^{∗}} Hom(H_{n}(G_{2}), H^{k}(B)).

By functoriality of Grothendieck’s classes c_{i} ([G,2.3]) we have

f^{∗}c_{i}(ρ_{1}) = c_{i}(f^{∗}ρ_{1}) = c_{i}(φ^{∗}ρ_{2}) = φ^{∗}c_{i}(ρ_{2}).

Hence f^{∗}Φc_{i}(ρ_{1}) = φ^{∗}Φc_{i}(ρ_{2}) as maps from H_{n}(GL_{N}(A); ℤ/q) to H_{et}^{k}(B; μ_{q}^{⊗i}). Letting N → ∞ and composing with the Hurewicz map yields the Proposition.

Gillet and Shekhtman used the same construction in [Gi] and [Sh], but with the integral Hurewicz map K_{n}(A) → H_{n}(GL(A); ℤ), to construct integral Chern classes

(2.2)  c_{ik} : K_{n}(A) → H_{et}^{k}(A; μ_{q}^{⊗i}),  n + k = 2i, n ≥ 0.

Example 2.2.1. (n = 1) By construction ([Sh, 1.b],[S, p. 279]), c_{11} is the homomorphism

c_{11} : K_{1}(A) ⟶^{det} A^{∗} ⟶ A^{∗}/A^{∗q} ⊂ H_{et}^{1}(A, μ_{q}).

All the other Chern classes c_{ik} : K_{1}(A) ⟶ H_{et}^{k}(A, μ_{q}^{⊗i}) vanish on A^{∗} ⊂ K_{1}(A). This is because the Hurewicz homomorphism maps A^{∗} into H_{1}(GL_{1}(A)), and c_{ik}(id)|H_{∗}(GL_{1}) is the map c_{ik}(id_{1}) arising via Φ from c_{i}(id_{1}). But since id_{1} is a 1-dimensional representation of GL_{1}(A), c_{i}(id_{1}) = 0 for i > 1.

Lemma 2.3. For n ≥ 2, the integral Chern classes (2.2) are the composition of Soulé’s Chern classes (2.1) with mod q reduction:

K_{n}(A) ⟶ K_{n}(A; ℤ/q) ⟶^{c_{ik}} H^{k}(A; μ_{q}^{⊗i})

The integral Chern classes are group homomorphisms for all n ≥ 0.

Proof. The first assertion follows from compatability of the two Hurewicz maps. The second results from the fact that the integral Hurewicz map is always a group homomorphism.

The mod q Chern classes (2.1) are not always homomorphisms, because the Hurewicz maps may not be. Here is the correct assertion, which follows immediately from theorem (1.2).

Proposition 2.4. Soulé’s Chern classes (2.1) are group homomorphisms if either q is odd or if q is even and n ≠ 2.

If q is even and n = 2 then Soulé’s Chern classes c_{ik} : K_{2}(A; ℤ/q) → H_{et}^{k}(A; μ_{q}^{⊗i}) satisfy:

c_{ik}(a + b) = c_{ik}(a) + c_{ik}(b) + \frac{q}{2}c_{ik}(\{∂a, ∂b\}).

Here a, b ∈ K_{2}(A; ℤ/q), i ≥ 1 and k = 2i − 2.

Corollary 2.5. (4|q). Let F be a ℤ[\frac{1}{2}]-algebra such that K_{1}(F) = F^{∗}. If 4|q then the Chern classes

c_{i,2i−2} : K_{2}(F; ℤ/q) ⟶ H_{et}^{2i−2}(F; ℤ/q)

are all group homomorphisms.

Proof. The correction term in (2.4) is zero by (1.4.1). Alternatively, this follows by applying c_{ik}(id) to (1.4.2).

Porism 2.5.1. If 4|q, the proof of 2.5 goes through to show that the Chern classes c_{i,2i−2} are all homomorphisms when restricted to the subgroup of K_{2}(A; ℤ/q) consisting of elements a such that ∂a ∈ μ_{q}(A).

We shall now describe the Chern classes on K_{2}(A; ℤ/2).

Observation 2.6. (n = 2) If A is a field (or more generally a semilocal ring), then the only nonzero Chern classes on K_{2}(A; ℤ/q) are

c_{10} : K_{2}(A; ℤ/q) ⟶ H^{0}(A; μ_{q}) = μ_{q}(A),
c_{22} : K_{2}(A; ℤ/q) ⟶ H^{2}(A; μ_{q}^{⊗2}).

This follows from the fact that K_{2}(A; ℤ/q) is a quotient of π_{2}(BGL_{2}(A)^{+}; ℤ/q), and c_{ik} on H_{∗}(GL_{2}) arises via Φ from c_{i}(id_{2}). Since id_{2} is a 2-dimensional representation, c_{i}(id_{2}) = 0 for i > 2. The same reasoning applies to K_{2}(ℤ; ℤ/q) = K_{2}(ℤ[\frac{1}{2}]; ℤ/q). Similar reasoning shows that if A is a field then c_{ik} vanishes on K_{n}(A; ℤ/q) unless n/2 ≤ i ≤ n. We will describe the Chern classes c_{10} and c_{22} in more detail below.

Example 2.6.1. (c_{10}) The Chern class map c_{10} is always a homomorphism — even when q = 2. Moreover, c_{10}(a) = 0 for every a ∈ K_{2}(A). Indeed, it follows easily from the argument on p. 279 of [S] that c_{10} is the Bockstein map, followed by the determinant map:

c_{10} : K_{2}(A; ℤ/q) ⟶^{∂} _{q}K_{1}(A) ⟶^{det} μ_{q}(A) = H_{et}^{0}(A; μ_{q}).

But since BGL(A)^{+} is an H-space, the Bockstein map ∂ : K_{2}(A; ℤ/q) → K_{1}(A) is a homomorphism.

Example 2.6.2. (c_{22}) Suppose that F is a field. By (2.5), c_{22} is a homomorphism when 4|q, so suppose that q = 2. By [Sh], c_{22} is the Galois symbol (up to sign) on the integral group K_{2}(F). For the real numbers ℝ, this implies that c_{22}({−1, −1}) = [ℍ], which is the nonzero element of H_{et}^{2}(ℝ; μ_{2}^{⊗2}) ≅ Br(ℝ) ≅ ℤ/2. Choose an element β ∈ K_{2}(ℝ; ℤ/2) such that ∂β = [−1] ∈ K_{1}(ℝ). It is well-known ([Wp, p. 396] [AT, p. 79]) that β generates K_{2}(ℝ; ℤ/2) ≅ ℤ/4 and that 2β is the Steinberg symbol {−1, −1}. By (2.4) we have c_{22}(β) ≠ c_{22}(−β), because

c_{22}(β) + c_{22}(−β) = c_{22}(0) − c_{22}({−1, −1}) = [ℍ] ≠ 0.

Note the resolution of the paradox mentioned in the introduction:

c_{22}(β + β) = 2c_{22}(β) + c_{22}({−1, −1}) = c_{22}({−1, −1}).

Example 2.6.3. Let SK_{2}(A; ℤ/q) denote the kernel of the map c_{10} : K_{2}(A; ℤ/q) → μ_{q}(A). There is an exact sequence, which is split if q ≢ 2 (mod 4):

0 → K_{2}(A) ⊗ ℤ/q → SK_{2}(A; ℤ/q) ⟶^{∂} _{q}SK_{1}(A) → 0.

Here_{q}SK_{1}(A) denotes the subgroup T or(SK_{1}(A), ℤ/q) of SK_{1}(A), which is in turn the kernel of det : K_{1}(A) → A^{∗}. The map c_{22} : SK_{2}(A; ℤ/q) → H_{et}^{2}(A; μ_{q}^{⊗2}) is a homomorphism for every A and every q, because the error term in 2.4 vanishes by the product formula for integral Chern classes [Sh] and 2.2.1:

c_{22}({∂a, ∂b}) = −c_{11}(∂a)c_{11}(∂b) = 0.

Definition 2.7 (Mod 2 Bott element). There are two generators of the cyclic group K_{2}(ℤ; ℤ/2) ≅ K_{2}(ℝ; ℤ/2) ≅ ℤ/4. Let β_{ℤ} be that generator of K_{2}(ℤ; ℤ/2) such that in H_{et}^{2}(ℝ; ℤ/2) ≅ ℤ/2 we have c_{22}(β_{ℝ}) = 0. This generator exists and is unique by (2.6.2). If A is any commutative ring containing 1/2, we write β = β_{A} for the image of β_{ℤ} in K_{2}(A; ℤ/2), and call it the mod 2 Bott element. Since H_{et}^{2}(ℤ[\frac{1}{2}]; ℤ/2) ≅ H_{et}^{2}(ℝ; ℤ/2), the observations of 2.6 imply that

c_{10}(β) = [−1] in H_{et}^{0}(A; ℤ/2) = μ_{2}(A)
c_{i,2i−2}(β) = 0  in H_{et}^{2i−2}(A; ℤ/2) for i ≠ 1

By construction, we have ∂β = [−1] in K_{1}(A) and 2β = {−1, −1} in K_{2}(A).

Remark 2.7.1. We warn the reader that c_{22}(−β) need not be zero, as illustrated in (2.6.2). Indeed, by 2.4 we have c_{22}(−β) = {−1, −1} ∈ H_{et}^{2}(A; ℤ/2). See Remark 4.2.2 below for a discussion of when this vanishes.

Definition 2.7.2 (Mod q Bott elements). We can generalize 2.7 as follows. Let ζ be a fixed primative q^{th} root of unity in a ring A. The mod q Bott element β ∈ K_{2}(A; ℤ/q) associated to ζ is the following element satisfying ∂β = ζ. Let μ_{q}(A) denote the cyclic group of q^{th} units in A and β_{0} ∈ π_{2}(Bμ_{q}(A); ℤ/q) the element corresponding to ζ under the isomorphism

π_{2}(Bμ_{q}(A); ℤ/q) ≅ π_{1}Bμ_{q}(A) ≅ μ_{q}(A).

Then β is the image of β_{0} under the natural map induced by Bμ_{q}(A) ⊆ BGL(A)^{+}. Since h_{2}(β) comes from H_{2}(BGL_{1}(A); ℤ/q) and c_{i}(id_{1}) = 0 for i > 0, we have c_{i,2i−2}(β) = 0 for i ≠ 1. If q = 2 this recovers the mod 2 Bott element. If q ≢ 2 (mod 4), it will follow from the product formula (3.1) or 3.2 (iv′) that

c_{i0}(β^{i}) = (−1)^{i−1}(i − 1)!ζ^{⊗i} in H_{et}^{0}(A; μ_{q}^{⊗i}) ≅ μ_{q}(A)^{⊗i}
c_{jk}(β^{i}) = 0  if j ≠ i.

The following well–known result will be needed in §5; special cases have been used in [S,IV.2], [Sus2,3.1] and [L,pp.330,335,340].

Proposition 2.8 (Soulé). If i, q and r are integers ≥ 1, then the cohomology sequence associated to 0 → μ_{q}^{⊗i} → μ_{qr}^{⊗i} → μ_{r}^{⊗i} → 0 is compatible with the Chern classes and the K-theory Bockstein, in the sense that the following diagram commutes for n ≥ 2.

K_{n}(A; ℤ/q) ⟶ K_{n}(A; ℤ/qr) ⟶ K_{n}(A; ℤ/r) ⟶^{∂} K_{n−1}(A; ℤ/q)
c_{ik}↓  ^{c_{ik}}↓  ^{c_{ik}}↓  ^{c_{i,k+1}}↓
H^{k}(A; μ_{q}^{⊗i}) ⟶ H^{k}(A; μ_{qr}^{⊗i}) ⟶ H^{k}(A; μ_{r}^{⊗i}) ⟶^{δ} H^{k+1}(A; μ_{q}^{⊗i})

Proof. ([S,IV.3.1]). Since the Hurewicz maps are compatible with the Bockstein [N,3.2.ii], we may replace K_{n}(A; ℤ/q) with H_{n}(G; ℤ/q) in order to prove compatability with Chern classes. If C_{•}(G) is the standard resolution of BG and Γ(L_{q}^{•}) is the complex of sections of a resolution of μ_{q}^{⊗i}, the naturality of Φ implies that the diagram

C_{•}(G) ⊗ ℤ/q ⟶ C_{•}(G) ⊗ ℤ/qr ⟶ C_{•}(G) ⊗ ℤ/r
Φ(c_{i})↓  ^{Φ(c_{i})}↓  ^{Φ(c_{i})}↓
Γ(L_{q}^{•})  ⟶  Γ(L_{qr}^{•})  ⟶  Γ(L_{r}^{•})

not only commutes in the derived category D(ℤ), but is a morphism of triangles. Passing to homology yields the Proposition.

3. The Product Formula.

In order to have a formula for c_{ik}({a, b}), one needs the K-theory product {a, b} to be defined in K_{∗}(A; ℤ/q). If q is odd or if 4|q then this product exists (see [Br, 1.6]). When q = 2, the mod 2 product {a, b} only exists if the integral product {∂a, ∂b} vanishes; see (1.6).

Suppose that a ∈ K_{m}(A; ℤ/q) and b ∈ K_{n}(A; ℤ/q) are such that the product {a, b} is defined. We assume that m, n ≥ 1, using the convention that K_{1}(A; ℤ/q) = K_{1}(A)/qK_{1}(A).

Soulé’s product formula states that

(3.1)  c_{ik}(\{a, b\}) = ∑_{i_{1}+i_{2}=i} \frac{−(i − 1)!}{(i_{1} − 1)! (i_{2} − 1)!} c_{i_{1}k_{1}}(a)c_{i_{2}k_{2}}(b).

Here the sum is over all i_{1} ≥ m/2 and i_{2} ≥ n/2 such that i_{1} + i_{2} = i, and we have set k = 2i − m − n, k_{1} = 2i_{1} − m, k_{2} = 2i_{2} − n.

Soulé proves in II.3 of [S] that his product formula (3.1) holds for odd q. Shekhtman points out in [Sh] that Soulé’s product formula (3.1) also holds for integral Chern classes, i.e., when both a and b come from integral K-theory. The purpose of this section is to analyze the situation when q is even.

Theorem 3.2. Suppose that q is even. Then Soulé’s product formula (3.1) holds for c_{ik}({a, b}) in the following cases:

(i) Neither m nor n is 2. (i′) If m = 2 then a ∈ K_{2}(A), and if n = 2 then b ∈ K_{2}(A). (ii) i = 2. That is, if k = 4 − m − n then in H_{et}^{k}(A; μ_{q}^{⊗2}) we have

c_{2k}({a, b}) = −c_{1,2−m}(a)c_{1,2−n}(b).

(iii) i = 3, n = 2 and m ≥ 3. That is, c_{3k}({a, b}) = −2c_{2,4−m}(a) ∪ c_{10}(b). (iv) (4|q) and K_{1}(A) = A^{∗}. (iv′) (4|q) and: if m = 2 then ∂a ∈ A^{∗}; if n = 2 then ∂b ∈ A^{∗}.

(v) i ≥ 5 and K_{1}(A) = A^{∗}. (v′) i ≥ 5 and: if m = 2 then ∂a ∈ A^{∗}; if n = 2 then ∂b ∈ A^{∗}.

Remark 3.2.1. As mentioned in the introduction, the only version of the product formula used in [Sus2] is when a ∈ K_{1}(A), b ∈ K_{2}(A; ℤ/q) and i = 2. Indeed, by 3.2(ii) we have

c_{21}({a, b}) = −c_{11}(a)c_{10}(b) = {det(a), det(∂b)}.

If q = 2, n = 2, and i = 3, 4 there is an extra term in the product formula. Note that if b ∈ K_{2}(A; ℤ/2) is such that ∂b is a unit, and \frac{1}{2} ∈ A, then ∂b = ±1 on each component of A. If ∂b = +1 then b ∈ K_{2}(A), and Soulé’s product formula holds by 3.2(i′). If ∂b = −1 then b differs from the mod 2 Bott element β of (2.7) by an element b′ of K_{2}(A). If m ≥ 1 then by 2.4 we have c_{ik}({a, b}) = c_{ik}({a, β})+c_{ik}({a, b′}). The last term is described by the product formula and the next Theorem describes c_{ik}({a, β}).

Theorem 3.3. Let β ∈ K_{2}(A; ℤ/2) be the mod 2 Bott element.

(i) Suppose that either a ∈ K_{m}(A) for m ≥ 1, or that a ∈ K_{m}(A; ℤ/2) for m ≥ 3 and the product {a, β} ∈ K_{m+2}(A; ℤ/2) is defined. Write [−1] for c_{10}(β) ∈ H^{0}(A; μ_{2}) = μ_{2}(A) and {−1, −1} ∈ H_{et}^{2}(A; ℤ/2) for the square of the class of −1 in A^{∗}/A^{∗2} ⊆ H_{et}^{1}(A; μ_{2}). Then in H_{et}^{k}(A; ℤ/2) we have:

c_{ik}\{a, β\} = (1 − i)c_{i−1,k}(a) ∪ [−1] + \binom{i − 1}{2} c_{i−2,k−2}(a) ∪ \{−1, −1\}.

The first term represents Soulé’s product formula (3.1). (ii) (a = β) Suppose that the product {β, β} is defined in K_{4}(A; ℤ/2), i.e., that

{−1, −1} = 0 in K_{2}(A). Then c_{22}({β, β}) = [−1] ⊗ [−1] ≠ 0, and all other Chern classes c_{ik}({β, β}) vanish.

Example 3.4. Theorem 3.3 corrects the assertion on p. 13 of [Sus2] – that Soulé’s product formula remains valid for q = 2^{ν}. For example, if β ∈ K_{2}(ℝ; ℤ/2) is the mod 2 Bott element, then we get the calculations in the introduction:

\begin{aligned}c_{33}(\{−1, β\}) & = \{−1, −1, −1\} ≠ 0 \\ c_{44}(\{−1, −1, β\}) & = \{−1, −1, −1, −1\} ≠ 0 \end{aligned}  \begin{matrix}in H_{et}^{3}(ℝ; ℤ/2) ≅ ℤ/2, \\ in H_{et}^{4}(ℝ; ℤ/2) ≅ ℤ/2. \end{matrix}

In both cases Soulé’s product formula (3.1) fails to hold.

(3.5.0). In order to prove Theorems 3.2 and 3.3, we shall modify the proof given by Soulé for odd q in [S, II.3]. For this, we need the maps

Φc_{I} : H_{n}(GL(A); ℤ/q) ⟶ H_{et}^{k}(A, μ_{q}^{⊗i})

associated to an ordered multi-index I = (i_{1}, …, i_{α}) with i = Σi_{γ} and n + k = 2i. The map Φc_{I} = Φc_{I}(id) is the composition of the diagonal map Δ_{∗} : H_{∗} ⟶ H_{∗} ⊗ ⋯ ⊗ H_{∗} with the tensor product of the maps Φ(c_{i_{γ}} (id)) constructed at the start of section 2.

We are going to apply the maps Φc_{I} to Hurewicz elements h_{n}(a). If I = (i) then Φc_{(i)}(h_{n}(a)) = c_{ik}(a), as expected. If h_{n}(a) is primitive and I has length ≥ 2 then Φc_{I}(h_{n}a) = 0 because all the Φ(c_{i}(id)) vanish on 1 ∈ H_{0}.

Lemma 3.5. Suppose that b ∈ K_{2}(A; ℤ/q). If I has length ≥ 3 then Φc_{I}(h_{n} b) = 0. If I = (i_{1}, i_{2}) and q is even, then

Φc_{I}(h_{2}(b)) = (\frac{q}{2})c_{i_{1}k_{1}}(∂b) ∪ c_{i_{2}k_{2}}(∂b), where k_{γ} = 2i_{γ} − 1.

Proof. This follows immediately from Proposition 1.1 and the fact that the h_{1}(∂b) are primitive elements of H_{1}.

Corollary 3.5.1. Suppose that b ∈ K_{2}(A; ℤ/q) and that u = ∂b is a unit of A. Then Φc_{(i_{1},i_{2})}(h_{2}b) = 0 unless both I = (1, 1) and q ≡ 2 (mod 4). In that case,

Φc_{(1,1)}(h_{2}b) = c_{11}(∂b)^{2} = {−1, u} ∈ H_{et}^{2}(A; μ_{2}^{⊗2}) = H_{et}^{2}(A; ℤ/2).

Proof. If i_{2} > 1 then c_{i_{2}k_{2}}(u) = 0 by (2.2.1). If 4|q then we have a multiple of 2c_{1k}^{2}(∂b), which vanishes in the graded ring H^{∗}. Finally, we identify c_{11}(∂b) with u ∈ A^{∗}/A^{∗2} ⊆ H^{1}(A; μ_{2}) by (2.2.1) and cite the well-known identity {u, u} = {−1, u} in H^{2}.

In [RRR], Grothendieck defined certain universal polynomials

Q_{i}(x_{1}, …, x_{i−1}; y_{1}, …, y_{i−1}) = ∑ a_{IJ} x^{I}y^{J},

in order to describe the Chern classes of a product of vector bundles. Here the notation is such that I and J are ordered multi-indices, and x^{I} means the product of the x_{i_{γ}} as i_{γ} runs through I. The polynomial Q_{i} is homogeneous of degree i, provided that x_{j} and y_{j} have degree j. The first few polynomials are:

Q_{1} = 0
(3.6)  Q_{2} = −x_{1}y_{1}
Q_{3} = (x_{1}^{2}y_{1} + x_{1}y_{1}^{2}) − 2(x_{1}y_{2} + x_{2}y_{1})
Q_{4} = −3(x_{1}y_{3} + x_{3}y_{1}) − 6x_{2}y_{2} + 3(x_{1}^{2}y_{2} + x_{2}y_{1}^{2})
+ 3(x_{1}x_{2}y_{1} + x_{1}y_{1}y_{2}) − (x_{1}^{3}y_{1} + x_{1}^{2}y_{1}^{2} + x_{1}y_{1}^{3}).

These formulas will be derived in 3.8 and 3.9 below. In general, we know from [RRR, (1.18)] that the coefficient of x_{m}y_{n} in Q_{m+n} is

(3.6.1)  a_{mn} = \frac{−(m + n − 1)!}{(m − 1)! (n − 1)!} .

Soulé uses the Q_{i} to define a map Φ_{i} = ΦQ_{i}(c_{∗}(id); c_{∗}(id)) by the formula Φ_{i} = ∑ a_{IJ} Φc_{I}(id) ∪ Φc_{J}(id) and observes (for odd q) that the diagram

K_{m}(A; ℤ/q) × K_{n}(A; ℤ/q) ⟶^{μ} K_{m+n}(A; ℤ/q)
h_{m}×h_{n}↓  ^{c_{i}}↓
H_{m}(GL(A)) ⊗ H_{n}(GL(A)) ⟶^{Φ_{i}} H_{et}^{•}(A; μ_{q}^{⊗i})

commutes. That is, we have:

(3.7)  c_{ik}({a, b}) = ∑ a_{IJ}Φc_{I}(h_{m}(a)) ∪ Φc_{J}(h_{n}(b)).

Soulé’s observation applies equally well if 4|q (when μ is defined), or even if q = 2, provided that we restrict to the subset π_{m} × π_{n} of K_{m}(A; ℤ/q) × K_{n}(A; ℤ/q) on which the product is defined by (1.6).

By (3.6.1), Soulé’s product formula (3.1) is the sum of those terms in (3.7) in which both I and J have length 1. By 3.5 we can ignore terms in (3.7) in which I or J has length ≥ 3. Therefore the deviation from the product formula is given by terms in which I or J have length 2, and these are described by 3.5.

Proof of 3.2. As long as the elements h_{m}a and h_{n}b are primitive, the only nonzero terms in (3.7) are the ones in the product formula (3.1). This is the case in (i) and (i′). The explicit formula for Q_{2} proves 3.2(ii). Part (iii) follows from 3.7.1 below, and part (iv) follows immediately from 3.5.1. Part (v) follows from Theorem 3.3, which we will prove below.

(3.7.1) Product formula for c_{3k}. The explicit formula for Q_{3} and (3.5) shows that if 4|q then Soulé’s product formula holds:

c_{3k}({a, b}) = −2c_{1k_{1}}(a)c_{2k_{2}}(b) − 2c_{2k_{2}}(a)c_{1k_{1}}(b).

When q = 2, things are more complicated. If we fix b ∈ K_{2}(A; ℤ/2), then c_{3k}({a, b}) is only defined for m ≤ 4. If a ∈ K_{1}(A), then

c_{33}({a, b}) = c_{11}(a)c_{11}^{2}(∂b) = [det(a)] ∪ [det(∂b)]^{2}.

This yields the formula c_{33}({a, β}) = {det(a), −1, −1} of Theorem 3.3(i) and 3.4.

Suppose that a ∈ K_{2}(A; ℤ/2), and {a, b} is defined, i.e., that {∂a, ∂b} = 0 in K_{2}(A). Then

c_{32}({a, b}) = c_{11}^{2}(∂a)c_{10}(b) + c_{10}(a)c_{11}^{2}(∂b).

This formula has two consequences of interest. On the one hand, if a ∈ K_{2}(A) this yields c_{32}({a, b}) = 0 for every b. On the other hand, whenever {b, b} is defined we have c_{32}({b, b}) = 0. This implies the case i = 3 of 3.3(ii).

Finally, suppose that a is in K_{3}(A; ℤ/2) or K_{4}(A; ℤ/2) and {a, b} is defined. The explicit formula for Q_{3} shows that c_{3k}({a, b}) = 0, and yields 3.2(iii).

(3.7.2) Product formula for c_{4k}. Suppose first that a ∈ A^{∗} and that b ∈ K_{2}(A; ℤ/q). By 2.2.1 the only nonzero Chern class on a is c_{11}(a) = [a]. Using 3.5, the explicit formula for Q_{4} shows that in H^{5}(A; μ_{q}^{⊗4}):

c_{45}(\{a, b\}) = −3[a] ∪ c_{34}(b) + 3\frac{q}{2}[a] ∪ [det(∂b)] ∪ c_{23}(∂b).

Soulé’s product formula (3.1) holds iff the second term vanishes. This will be the case whenever ∂b is a unit, for then c_{23}(∂b) = 0 by 2.2.1.

Next, suppose that a ∈ K_{2}(A) and β is the mod 2 Bott element of K_{2}(A; ℤ/2). The explicit formula for Q_{4} shows that if 4|q then the product formula (3.1) holds:

c_{44}({a, β}) = −3c_{34}(a) ∪ [−1],

while if q = 2 we have:

c_{44}({a, β}) = c_{34}(a) ∪ [−1] + c_{22}(a) ∪ {−1, −1}.

In particular, if x, y ∈ A^{∗} then c_{34}({x, y}) = 0 by 2.6, so

c_{44}({x, y, β} = {x, y, −1, −1} in H^{4}(A; ℤ/2).

Finally, suppose that {β, β} is defined in K_{4}(A; ℤ/2). Then the explicit formula for Q_{4} yields the case i = 4 of 3.3(ii):

c_{44}({β, β}) = −c_{11}(∂β)^{2} = c_{22}({−1, −1}) = 0.

For the proof of the next result, which is used in the proof of Theorem 3.3 (and hence in the proof of 3.2), we shall need a few observations about symmetric polynomials. Formally factoring 1 + xt^{m} = ∏(1 + α_{i}t), note that the symmetric polynomials σ_{i} in the α_{i}’s are: σ_{i} = 0 for i ≠ m, and σ_{m} = x. Setting s_{k} = ∑ α_{i}^{k}, Newton’s formula (see [CC, p. 195]) is:

s_{n} − σ_{1}s_{n−1} + ⋯ + (−1)^{n}nσ_{n} = 0.

This easily yields s_{k} = 0 for k < m, s_{m} = (−1)^{m−1}mx, and s_{k} = 0 if k > m and m ∤ k. By induction, we also see that if k = jm then s_{jm} = (−1)^{j(m−1)}mx^{j}. With these preliminaries out of the way, we are ready to proceed.

Proposition 3.8. The coefficient of x_{m}y_{1}^{n} in Q_{m+n}(x_{1}, …; y_{1}, …) is

a_{m,(1,...,1)} = (−1)^{n} \binom{m + n − 1}{n} .

In particular, the coefficient of x_{m}y_{1}^{2} in Q_{m+2} is a_{m,(1,1)} = ({m+1 \atop 2}).

Proof. It suffices to evaluate Q_{i} = Q_{i}(0, …, x, 0; y, 0, …) modulo x^{2}. According to (1.17 bis) of [RRR], Q_{i} is the coefficient of t^{i} in the formal expansion of

(1 + xt^{m}) ∗ (1 + yt) = ∏_{i=1}^{m} \frac{(1 + (α_{i} + y)t)}{(1 + α_{i}t)(1 + yt)}.

The logarithm of this power series is

L = ∑_{i,k} \frac{(−1)}{k}^{k−1}(α_{i} + y)^{k}t^{k} − ∑_{i,k} \frac{(−1)}{k}^{k−1}α_{i}^{k}t^{k} − m ∑_{k} \frac{(−1)}{k}^{k−1}y^{k}t^{k}.

The above remarks about the s_{k} show that the leading term of L is of degree m+1, namely t^{m+1} times:

Q_{m+1} = \frac{(−1)^{m}}{m+1} \{∑_{j}(α_{i}+y)^{m+1} − s_{m+1} − my^{m+1}\} = (−1)^{m}s_{m}y = −mxy.

This is no surprise, as it recovers Grothendieck’s observation in (1.18) of [RRR] that Q_{m+1} = −mxy. We are interested in the term of degree m + n in t:

(3.8.1) \begin{aligned}\frac{(−1)^{m+n−1}}{m+n} & \{∑(α_{i} + y)^{m+n} − s_{m+n} − my^{m+n}\} t^{m+n} \\ & = \frac{(−1)^{m+n−1}}{m+n} ∑_{k=0}^{m+n} \binom{m+n}{k} s_{k}y^{m+n−k}t^{m+n}. \end{aligned}

Modulo x^{2}, only the term k = m is nonzero, and this equals

(−1)^{n} \frac{m}{m + n} \binom{m+n}{n} xy^{n}t^{m+n} = (−1)^{n} \binom{m + n − 1}{n} xy^{n}t^{m+n}.

Therefore modulo x^{2} we have:

(1 + xt^{m}) ∗ (1 + yt) = exp(−mxyt^{m+1} + \binom{m + 1}{2} xy^{2}t^{m+2} − ⋯)

= 1 − mxyt^{m+1} + \binom{m + 1}{2} xy^{2}t^{m+2} − · · · .

It follows that Q_{m+n}(0, …, x, 0; y, …) ≡ (−1)^{n} \binom{m+n−1}{n} xy^{n}t^{m+n}, as required.

Exercise 3.8.2. Use (3.8.1) to show that the coefficient of x_{1}x_{2}y_{1} in Q_{4} is 3.

Proof of Theorem 3.3. Let β ∈ K_{2}(A; ℤ/2) be the mod 2 Bott element (see 2.7). By 3.5.1, the only nonzero terms in (3.7) in which J has length 2 are those with J = (1, 1), and have the form a_{IJ}Φc_{I}(h_{m}a) ∪ {−1, −1}. To prove (i), when h_{m}(a) is primitive, we only need the coefficients a_{iJ}, which equal ({i−1 \atop 2}) by 3.8.

To prove (ii), suppose that {β, β} is defined in K_{4}(A; ℤ/2). By 3.2(ii) we have c_{20}({β, β}) = −c_{10}(β)^{2}. This is the nonzero element of H_{et}^{0}(A; μ_{2}^{⊗2}) ≅ μ_{2} ⊗ μ_{2}. We saw in (3.7.1) and (3.7.2) that c_{32} and c_{44} also vanish on {β, β}. For i ≥ 5 we argue as follows. By 2.7, c_{ik}(β) = 0 for i ≠ 1. By 3.5 and 3.5.1 this means that Φc_{I}(h_{2}β) = 0 unless I is (1), (2) or (1, 1). These belong to H^{k}(A; ℤ/2) for k = 0, 1 and 2, respectively. Since c_{ik}({β, β}) belongs to H^{2i−4}(A; ℤ/2), the right-hand side of (3.7) must be zero for dimensional reasons.

Lemma 3.9. The coefficient of x_{1}^{i}y_{1}^{m−i} in Q_{m}(x_{1}, ...; y_{1}, ...) is: zero if i = 0, m and (−1)^{m−1} when 1 ≤ i < m.

Proof. According to (1.17bis) of [RRR], Q_{m}(x, 0, ...; y, 0, ...) is the coefficient of t^{m} in the formal power series

(1 + xt) ∗ (1 + yt) = \frac{(1 + (x + y)t)}{(1 + xt)(1 + yt)} = (1 + (x + y)t) ∑(−xt)^{i} ∑(−yt)^{j}.

Hence Q_{m} = (−1)^{m−1}(x^{m−1}y + ⋯ + xy^{m−1}).

Example 3.9.1. The explicit formulas (3.6) for the homogeneous polynomials Q_{3} and Q_{4} follow from (3.6.1), 3.8, 3.8.2 and the following cases of 3.9:

Q_{3}(x, 0; y, 0) = x^{2}y + xy^{2}
Q_{4}(x, 0, 0; y, 0, 0) = −(x^{3}y + x^{2}y^{2} + xy^{3}).

We conclude this section with a product formula for c_{ik}({a, b}) when ±a ∈ K̃_{0}(A) is the reduced class [L] − 1 of a line bundle L. By definition, c_{12}(a) is the class of L^{±1} in P ic(A)/q ⊆ H_{et}^{2}(A; μ_{q}) and c_{i,2i}(a) = 0 for i ≠ 1. Write λ for c_{12}(a) ∈ P ic(A)/q ⊆ H^{2}(A; μ_{q}), and consider the following product formula:

(3.10)  c_{ik}(\{a, b\}) = ∑_{i_{1}+i_{2}=i}(−1)^{i_{1}} \binom{i − 1}{i_{1}} λ^{i_{1}}c_{i_{2}k_{2}}(b).

The sum here is over all i_{1} ≥ 1 and i_{2} ≥ n/2 such that i_{1} + i_{2} = i, and we have set k = 2i − n, k_{2} = 2i_{2} − n.

Soulé gave a slightly different (incorrect) product formula in II.3 of [S], differing from (3.10) by ±i_{1}. We have expanded his rather terse proof in order to show the key role played by Lemma 3.8 above. Soulé has remarked that Suslin has also noticed the formula (3.10) (personal communication).

Remark 3.10.1. We must have rank(a) = 0 in (3.10). For example, 3.10 fails for a = 1: {1, b} = b but the right-hand side of (3.10) vanishes as c_{i,2i}(1) = 0 for all i. This corrects a typo in [S, II.3].

Theorem 3.11. Suppose that ±a ∈ K̃_{0}(A) is the reduced class [L]−1 of a line bundle, and that b ∈ K_{n}(A; ℤ/q). Then the product formula (3.10) holds for c_{ik}({a, b}) in the following cases:

(i) If n ≠ 2, or if q is odd. (ii) If n = 2 and b ∈ K_{2}(A). (iii) If n = 2, 4|q and ∂b ∈ A^{∗}. (iv) If n = 2 and i = 2, when (3.10) becomes

c_{22}({a, b}) = −λ ∪ [∂b].

If b ∈ K_{2}(A; ℤ/2), ∂b = −1 ∈ A^{∗} and i ≥ 3 then there is an extra term:

(3.11.1) c_{i,2i−2}\{a, b\} = ∑(−1)^{i_{1}} \binom{i − 1}{i_{1}} λ^{i_{1}}c_{i_{2}k_{2}}(b) + (−1)^{i−1}λ^{i−2} ∪ \{−1, −1\}.

Proof. If P is a finitely generated projective A-module, let 1_{P} denote the trivial representation of G in Aut(P) ⊆ GL(A). The corresponding homomorphisms

c_{ik}(1_{P}) : H_{n}(G; ℤ/q) → H_{et}^{k}(A; μ_{q}^{⊗i})

are zero unless n = 0 and k = 2i, when 1 ∈ H_{0}(G; ℤ/q) maps to c_{i,2i}(P). As in definition (3.5.0), if J = (j_{1}, ...) is a multi-index let Φc_{J}(1_{P}) denote the composition of the diagonal map Δ : H_{n} → H_{∗} ⊗ ⋯ ⊗ H_{∗} with the tensor product of the maps Φ(c_{j_{γ}} (1_{P})). Clearly this map is zero on H_{n}(G; ℤ/q) unless n = 0, when Φc_{J}(1_{P}) maps 1 to ∏ c_{j_{γ},2j_{γ}} (P). With this notation, Soulé proves on p. 265 of [S] that if a = [P] − rank(P) then

c_{ik}({a, b}) = c_{ik}(id_{m} ⊗ 1_{P})(h_{n}b)

= ΦQ_{i}(c_{∗}(id_{m}); c_{∗}(1_{P}))(Δh_{n}b) (3.11.2)

= Σa_{IJ}Φc_{I}(h_{n}b) ∪ c_{J}(1_{P})(1)

If the c_{i,2i}(P) vanish for i ≠ 1, for example if P is a line bundle, we can restrict to the terms in which J is the multi-index (1, ..., 1) of length i_{1}. If I = (i_{2}) and i_{1} + i_{2} = i, Lemma 3.8 shows that the corresponding term is

(−1)^{i_{1}} \binom{i − 1}{i_{1}} λ^{i_{1}}c_{i_{2}k_{2}}(b).

The sum of all these terms is the product formula (3.10). Therefore the formula (3.10) holds whenever h_{n}b is primitive. This proves (i) and (ii); (iv) follows from the explicit formula (3.6) for Q_{2}.

Now suppose that b ∈ K_{2}(A; ℤ/q) and ∂b is a unit of A. By 3.5.1 there are no more terms in (3.11.2) if 4|q, proving (iii). If q = 2 there is an extra term, when I = (1, 1) and J = (1, ..., 1). In this case we know by 3.9 that a_{IJ} = (−1)^{i−1}, so by 3.5.1 the extra term is ±λ^{i−2} ∪ {−1, ∂b}. This establishes formula (3.11.1).

Remark 3.11.3. As the proof shows, Theorem 3.11 also applies to any a ∈ K̃_{0}(A) such that c_{i,2i}(a) = 0 for i ≠ 1.

4. Localization and Transfer.

In this section, A will always denote a Dedekind domain, and F will denote its field of fractions. The residue fields of A will be written as k_{v}; if A is a discrete valuation ring (DVR) we will drop the subscript v and just write k.

All the preliminary material in III.1 of [S] on étale cohomology remains true if q = 2^{ν}, except for the comments on cohomological dimension of a global field at the end of III.1.1 and III.1.3. In particular, the localization sequences for a Dedekind domain A are valid for all q with \frac{1}{q} ∈ A:

(4.0) \begin{matrix}⋯ K_{n}(A; ℤ/q) ⟶ K_{n}(F; ℤ/q) ⟶^{∂} & ⊕_{v} K_{n−1}(k_{v}; ℤ/q) ⟶ K_{n−1}(A; ℤ/q) ⋯ \\ 0 ⟶ H_{et}^{1}(A, μ_{q}^{⊗i}) ⟶ H_{et}^{1}(F, μ_{q}^{⊗i}) ⟶^{∂} & ⊕_{v} H_{et}^{0}(k_{v}, μ_{q}^{⊗i−1}) ⟶ H_{et}^{2}(A, μ_{q}^{⊗i}) ⋯ \end{matrix}

The following result summarizes Propositions 2(i), 3 and 4 of [S,III.2-3]; case (a) is due to Quillen. Soulé’s proof goes through for all q, although in the proof of (c) we need to observe that condition (iii) on p. 27 of [HS] holds when q = 2^{ν}.

Proposition 4.1. Suppose one of the following hypotheses holds:

(a) A = k[t];

(b) A is a DVR and char(F) = char(k);

(c) A is a complete DVR and k is a finite field. Then the localization sequences (4.0) for the mod q K-theory and étale cohomology of A and F split up into short exact sequences:

0 ⟶ K_{n}(A; ℤ/q) ⟶ K_{n}(F; ℤ/q) ⟶^{∂} ⊕_{v}K_{n−1}(k_{v}; ℤ/q) ⟶ 0

0 ⟶ H_{et}^{k}(A; μ_{q}^{⊗i}) ⟶ H_{et}^{k}(F; μ_{q}^{⊗i}) ⟶^{∂} ⊕_{v}H_{et}^{k−1}(k_{v}; μ_{q}^{⊗i}) ⟶ 0.

In case (a) the Weil reciprocity law ΣN_{v}∂_{v} + ∂_{∞} = 0 also holds, where ∂_{∞} corresponds to the place at infinity for F = k(t). In case (c) the map K_{n}(A; ℤ/q) → K_{n}(k; ℤ/q) is a split surjection.

The next result concerns the compatability of these sequences with Chern classes. Consider the (noncommutative) diagrams arising from (4.0):

K_{n}(A; ℤ/q) ⟶ K_{n}(F; ℤ/q) ⟶^{∂} ⊕K_{n−1}(k_{v}; ℤ/q)
c_{ik}↓  ^{c_{ik}}↓  ↓^{c_{i−1,k−1}}
H_{et}^{k}(A; μ_{q}^{⊗i}) ⟶ H_{et}^{k}(F; μ_{q}^{⊗i}) ⟶^{∂} ⊕H_{et}^{k−1}(k_{v}; μ_{q}^{⊗i})

Proposition 4.2. Suppose that A is a Dedekind domain containing \frac{1}{q}. Then with one exception, the Chern classes c_{ik} on K_{n}(F; ℤ/q) satisfy

∂c_{ik} = (1 − i)c_{i−1,k−1}∂.

This exception is when q = 2, n = 3, c_{ik} = c_{33} and ℚ ⊆ A. In this case, the formula ∂c_{33} = −2c_{22}∂ = 0 holds iff {−1, −1} = 0 in every H_{et}^{2}(k_{v}; ℤ/2).

Proof. (Cf. pp. 271 and 274 of [S].) By naturality, we may assume that A is a complete DVR with parameter π. Now K_{∗}(A; ℤ/q) ≅ K_{∗}(k; ℤ/q) by [Gab]. Given b ∈ K_{n−1}(A; ℤ/q), form {π, b} ∈ K_{n}(F; ℤ/q) and note that ∂{π, b} is the image b̄ of b in K_{n−1}(k; ℤ/q). Unless q = 2, n = 3 and i = 3, Soulé’s product formula (3.1) applies by 3.2(i,ii,iv,v) and 3.7.2, yielding

c_{ik}{π, b} = (1 − i)c_{11}(π)c_{i−1,k−1}(b),
∂c_{ik}{π, b} = (1 − i)c_{i−1,k−1}(b̄).

By a diagram chase we get ∂c_{ik} = (1 − i)c_{i−1,k−1}∂.

In the exceptional case (q = 2, n = i = 3), we argue as follows. Given b ∈ K_{2}(A; ℤ/2), the product formula (3.7.1) yields

\begin{aligned}c_{33}\{π, b\} & = c_{11}(π)c_{11}^{2}(∂b) \\ ∂c_{33}\{π, b\} & = c_{11}^{2}(∂b̄) \end{aligned}  \begin{matrix}in H_{et}^{3}(F, ℤ/2), and \\ in H_{et}^{2}(k, ℤ/2). \end{matrix}

Since ∂b̄ ∈ μ_{2}(k) = {±1}, we see that formula (4.2) holds (∂c_{33} = 0) iff {−1, −1} = 0 in H_{et}^{2}(k; ℤ/2). This is impossible if char(k) ≠ 0.

Remark 4.2.1. Levine uses this proposition for c_{21} in the proof of Theorem 4.5 and 4.12 of [L]. For F = k(t) the following diagram commutes, even if q = 2.

K_{3}(F; ℤ/q) ⟶^{∂} ⊕_{x}K_{2}(k(x); ℤ/2) ⟶ 0
c_{21}↓  ^{−c_{10}}↓
H′(F; μ_{q}^{⊗2}) ⟶^{∂} ⊕_{x}H_{et}^{0}(k(x); μ_{q}) ⟶ 0

Remark 4.2.2. The symbol {−1, −1} ∈ H_{et}^{2}(k; ℤ/2) is nonzero for k = ℝ as well as for k = ℚ̂_{2} and any extension of ℚ̂_{2} of odd degree. It is zero for every other local field. Therefore if k is a number field, this symbol is nonzero if and only if either k has a real embedding, or there is an unramified prime 𝔭 over 2 with residue field 𝔽_{2^{f}}, f odd. This observation is due to W. Raskind.

Proposition 4.3. Suppose that A is a Dedekind domain containing \frac{1}{q} and a primitive q^{th} root of unity ζ_{q}, q ≢ 2(mod 4). Then for every finite residue field k = A/𝔭 the following diagrams commute for all i ≥ 2, with n and k so that n + k = 2i.

K_{n}(k; ℤ/q) ⟶^{τ} K_{n}(A; ℤ/q)
↓^{(1−i)c_{i−1,k−2}}  ↓^{c_{ik}}
H_{et}^{k−2}(k; μ_{q}^{⊗i−1}) ⟶^{τ} H_{et}^{k}(A; μ_{q}^{⊗i})

Proof. By [Br, 2.6] and (3.1), every element b̄ of K_{n}(k; ℤ/q) lifts to an element b of K_{n}(A; ℤ/q) with the property that c_{jk}(b) = 0 unless k ≤ 1. Now [k] = 1 − [𝔭] in K_{0}(A), so if we set λ = c_{12}([k]) ∈ H^{2}(A; μ_{q}) then the product formula 3.11(i,iii) yields:

c_{i,k}τ(b̄) = c_{i,k}({[k], b}) = −(i − 1)λ ∪ c_{i−1,k−2}(b) = (1 − i)τc_{i−1,k−2}(b̄).

Corollary 4.3.1. Suppose that A is a Dedekind domain containing \frac{1}{q} and ζ_{q}, q ≢ 2(mod 4). If all the residue fields k_{v} of A are finite, there is a map of localization sequences (4.0):

⋯ K_{n+1}(F; ℤ/q) ⟶^{∂} ⊕K_{n}(k_{v}; ℤ/q) ⟶^{τ} K_{n}(A; ℤ/q) ⟶K_{n}(F; ℤ/q) ⟶^{∂} ⋯
↓^{c_{i,k−1}}  ↓^{(1−i)c_{i−1,k−2}}  ↓^{c_{ik}}  ↓^{c_{ik}}
H_{et}^{k−1}(F; μ_{q}^{⊗i}) ⟶^{∂} ⊕H_{et}^{k−2}(k_{v}; μ_{q}^{⊗i−1}) ⟶ H_{et}^{k}(A; μ_{q}^{⊗i}) ⟶H_{et}^{k}(F; μ_{q}^{⊗i}) ⟶^{∂} ⋯

Proof. Combine 2.1.1, 4.2 and 4.3.

Remark 4.3.2. See (5.3.1) for a version with q = 2 and i = 2. Clearly, the result (4.3.1) holds in much greater generality. For example if A is the coordinate ring of a smooth affine curve over an algebraically closed field then 4.3 and 4.3.1 hold for all q. However, care must be taken when q = n = 2, as the following example shows.

Example 4.3.3. When A = ℝ[x, y]/(x^{2} + y^{2} = 1) and k = A/(x − cos θ, y − sin θ) is any residue field isomorphic to ℝ, the following diagram does \underline{not} commute.

K_{2}(ℝ; ℤ/2) ⟶^{τ} K_{2}(A; ℤ/2)
↓^{−2c_{22}=0}  ↓^{c_{34}}
H_{et}^{2}(ℝ; μ_{2}^{⊗2}) ⟶^{τ} H_{et}^{4}(A; μ_{2}^{⊗3})

Indeed, λ = c_{12}([k]) is the nonzero element of P ic(A) ≅ ℤ/2 in H_{et}^{2}(A; μ_{2}). By [PW, 1.11] the cup product with {−1, −1} induces an isomorphism H_{et}^{2}(A; μ_{2}) ≅ H_{et}^{4}(A; μ_{2}^{⊗3}) ≅ ℤ/2 ⊕ ℤ/2. Letting β denote the mod 2 Bott element, the product formula (3.11.1) yields

c_{34}τ(β) = c_{34}({[k], β}) = λ ∪ {−1, −1} ≠ 0.

We remark that by [RW, 2.1] we have K_{2}(A; ℤ/2) ≅ K_{2}(ℝ; ℤ/2) ⊕ ℤ/2, the last summand being generated by τ(β). Since c_{22}τ(β) = −λ ∪ c_{10}(β) by 3.11(iv), it follows that c_{22} : SK_{2}(A; ℤ/2) ≅ H_{et}^{2}(A; ℤ/2) ≅ ℤ/2 ⊕ ℤ/2. This isomorphism should be compared with 5.3 below.

Proposition 4.4. Let k ⊆ ℓ be a finite separable extension of fields containing 1/q. Let N denote the transfer map in either K-theory or étale cohomology, and fix x ∈ K_{n}(ℓ; ℤ/q), i and k = 2i − n. Then

i{c_{ik}(Nx) − Nc_{ik}(x)} = 0.

Proof. Following the proof of [S, Thm. 2], choose a residue field k_{w} of A = k[t] isomorphic to ℓ. By 4.1(a) there is a y ∈ K_{n+1}(k(t); ℤ/q) mapping to x · 1_{ℓ} in ⊕K_{n}(k_{v}; ℤ/q), and Weil reciprocity shows that ∂_{∞}(y) = −Nx. Writing ∂_{v} for the summands of the cohomology boundary map in (4.0), we see by 4.2 that

∂_{v}c_{i+1,k+1}(y) = \{ \begin{matrix}−ic_{ik}(x), & if v = w \\ +ic_{ik}(Nx), & if v = ∞ \\ 0 & otherwise \end{matrix}

unless q = 2, n = 2 and i = 2 (in which case the conclusion is trivial). Weil reciprocity for étale cohomology (4.1.a) finishes the proof.

Theorem 4.5. Let A be a Dedekind ring whose field of fractions F is a global field. Then for all \underline{even} n and all q the boundary map ∂ in the localization sequence for mod q K-theory is a surjection.

K_{n}(F; ℤ/q) ⟶^{∂} ⊕_{v}K_{n−1}(k_{v}; ℤ/q) → 0.

Proof. Soulé proves this for odd q in [S], Theorem 3; his argument remains valid when 4|q because in that case K_{∗}(A; ℤ/q) still has a product ([S2,p.327]). The case q = 4 implies the case q = 2 via the commutative diagram

K_{n}(F; ℤ/4) ⟶ ⊕K_{n−1}(k_{v}; ℤ/4)
↓  ↓
K_{n}(F; ℤ/2) ⟶ ⊕K_{n−1}(k_{v}; ℤ/2),

because when n is even the right vertical map is onto by (1.6.2).

Remark 4.5.1. The proof of 4.5 remains valid if A is any Dedekind domain for which all the residue fields are finite, SK_{1}(A) = 0 and SK_{1}(B_{q}) = 0, where B_{q} is the integral closure of A[ζ_{q}] in F (ζ_{q}), ζ_{q} a qth root of unity. I learned this from Clay Sherman [Sherm].

The following theorem is due to Soulé ([S,Thm 1], [S2,Thm 3], [Sherm]); I have included the short proof for the convenience of the reader.

Theorem 4.6. Let A be a Dedekind ring whose field of fractions F is a global field. Then the localization sequence for K-theory breaks up into exact sequences:

0 ⟶ K_{n}(A) ⟶ K_{n}(F) ⟶ ⊕_{v}K_{n−1}(k_{v}) ⟶ 0.

In particular, if n is odd and n ≠ 1 then K_{n}(A) ≅ K_{n}(F).

Proof. Let SK_{n}(A) denote the kernel of K_{n}(A) ⟶ K_{n}(F); it suffices to prove that SK_{n}(A) = 0 for all n. From Quillen’s computation of K_{∗}(𝔽_{t}), SK_{n}(A) is zero if n is even and finite if n is odd. If ℓ is a prime and S ≠ 0 is the ℓ-Sylow subgroup of SK_{n}(A), choose q = ℓ^{ν} annihilating S. From the diagram

K_{n}(A) ⟶^{q} K_{n}(A) ⟶ K_{n}(A; ℤ/q)
↓  ↓
K_{n}(F) ⟶ K_{n}(F; ℤ/q)

we see that S maps trivially into K_{n}(F) and nontrivially into K_{n}(A; ℤ/q). This contradicts Theorem 4.5 because n is odd.

Remark 4.6.1. As in (4.5.1), it suffices to assume that the residue fields k_{v} are finite and that for every q the integral closure B_{q} of A[ζ_{q}] satisfies SK_{1}(B_{q}) = 0.

5. The image of the Chern classes.

It is unrealistic to expect 2-primary Chern classes c_{ik} to be onto unless i ≤ 2. This is illustrated by any finite field 𝔽 of order r. By Quillen, we have

K_{2i}(𝔽; ℤ/q) ≅ K_{2i−1}(𝔽; ℤ/q) ≅ K_{2i−1}(𝔽) ⊗ ℤ/q ≅ ℤ/(q, r^{i} − 1).

As observed in III.1.3 of [S], these groups are abstractly isomorphic to

H_{et}^{0}(𝔽; μ_{q}^{⊗i}) ≅ H_{et}^{1}(𝔽; μ_{q}^{⊗i}) ≅ ℤ/(q, r^{i} − 1).

Proposition 5.1. Let 𝔽 be a finite field of order r. The maps c_{i0} and c_{i1} may be identified with multiplication by ±(i − 1)! on the cyclic group ℤ/(q, r^{i} − 1).

Proof ([S, IV.2]). First, suppose that 𝔽 contains a primitive q^{th} root of unity ζ. If β ∈ K_{2}(𝔽; ℤ/q) is such that ∂β = ζ then β^{i} and ζβ^{i−1} generate K_{2i}(𝔽; ℤ/q) and K_{2i−1}(𝔽; ℤ/q). On the other hand H_{et}^{0}(𝔽; μ_{q}^{⊗i}) ≅ μ_{q}^{⊗i} is generated by ζ^{⊗i} and H_{et}^{1}(𝔽; μ_{q}^{⊗i}) is generated by c_{11}(ζ) ⊗ ζ^{⊗i−1}. Therefore it suffices to observe that the product formula (3.1), which is valid by 3.2(iv′) and 3.3(ii), yields:

c_{i0}(β^{i}) = (−1)^{i−1}(i − 1)!ζ^{⊗i},
c_{i1}(ζβ^{i−1}) = (−1)^{i−1}(i − 1)!c_{11}(ζ) ⊗ ζ^{⊗i−1}.

If 𝔽 does not contain ζ, we argue as follows. The inclusion 𝔽 ⊆ 𝔽(ζ) induces inclusions of K_{2i}(𝔽; ℤ/q) and H^{0}(𝔽; μ_{q}^{⊗i}) into K_{2i}(𝔽(ζ); ℤ/q) and H^{0}(𝔽(ζ); μ_{q}^{⊗i}). By naturality, c_{i0} = (−1)^{i−1}(i − 1)!. This implies the same result for c_{i1}, for if we set s = r^{i} − 1 then by 2.8 there is a commutative diagram:

K_{2i}(𝔽; ℤ/s) ⟶^{∂} K_{2i−1}(𝔽; ℤ/q) → 0
↓^{c_{i0}}  ↓^{c_{i1}}
H^{0}(𝔽; μ_{s}^{⊗i}) ⟶^{δ} H^{1}(𝔽; μ_{q}^{⊗i}) → 0.

We have already described the Chern classes c_{11} and c_{10} in 2.2.1 and 2.6.1. We now turn to a description of the classes c_{2k}.

If F is a field, the Murkurjev-Suslin Theorem of [MS2] states that c_{22} induces an isomorphism K_{2}(F) ⊗ ℤ/q → H_{et}^{2}(A; μ_{q}^{⊗2}) for all q. More recently, Levine [L] and Merkurjev-Suslin [MS] proved that for any field F the map c_{21} induces an exact sequence for all q:

K_{3}^{M}(F) → K_{3}(F; ℤ/q) →^{c_{21}} H_{et}^{1}(F; μ_{q}^{⊗2}) → 0.

Corollary 5.1.1. If A is the ring of integers in a local field F with residue field k, and \frac{1}{q} ∈ A, then K_{∗}(A; ℤ/q) ≅ K_{∗}(k; ℤ/q) by [Gab] and H_{et}^{∗}(A; μ_{q}^{⊗i}) ≅ H_{et}^{∗}(k; μ_{q}^{⊗i}), so again c_{i0} and c_{i1} may be identified with multiplication by ±(i − 1)!.

The groups K_{2i−1}(F; ℤ/q) and H_{et}^{1}(F; μ_{q}^{⊗i}) are also abstractly isomorphic by 4.1(c), and under this isomorphism we can again identify c_{i1} with multiplication by ±(i − 1)!.

Proposition 5.2. (Soulé [S,IV.1.5]). Let F be a field containing \frac{1}{q} and a primitive q^{th} root of unity ζ, q ≢ 2 (mod 4). Then for i ≥ 2 the cokernels of the maps

c_{i0} : K_{2i}(F; ℤ/q) → H_{et}^{0}(F; μ_{q}^{⊗i}) ≅ μ_{q}^{⊗i}
c_{i1} : K_{2i−1}(F; ℤ/q) → H_{et}^{1}(F; μ_{q}^{⊗i}) ≅ F^{∗} ⊗ μ_{q}^{⊗i−1}
c_{i2} : K_{2i−2}(F; ℤ/q) → H_{et}^{2}(F; μ_{q}^{⊗i}) ≅ _{q}Br(F) ⊗ μ_{q}^{⊗i−1}

have exponents dividing (i − 1)!.

Proof. Let β ∈ K_{2}(F; ℤ/q) be the mod q Bott element associated to ζ, as in 2.7.2. Let x ∈ K_{1}(F) and y ∈ K_{2}(F). The product formula (3.1), which holds by 3.2(iv′), yields

c_{i0}(β^{i}) = (−1)^{i−1}(i − 1)!ζ^{⊗i}
c_{i1}({x, β^{i−1}}) = (−1)^{i−1}(i − 1)!c_{11}(x) ⊗ ζ^{⊗i−1}
c_{i2}({y, β^{i−2}}) = (−1)^{i}(i − 1)!c_{22}(y) ⊗ ζ^{⊗i−2}.

Since c_{11} and c_{22} are onto by 2.2.1 and [MS2], the result follows.

Remark 5.2.1. If q = 2 (ζ = −1) the assertions of 5.2 are vacuous unless i = 2. We know that c_{22} is onto by [MS2], and c_{21} is onto since c_{21}({x, β}) = c_{11}(x)⊗ζ ([MS], [L]). But the conclusion of 5.2 fails for c_{20} since c_{20} need not be onto. If F = ℝ the map c_{20} : K_{4}(ℝ; ℤ/2) → H_{et}^{0}(ℝ; μ_{q}^{⊗2}) ≅ μ_{2}^{⊗2} is zero because c_{20}({−1, −1, β}) = 0 by 3.2(ii) and (2.6.1) and {−1, −1, β} is the only nonzero element of K_{4}(ℝ; ℤ/2) ≅ ℤ/2. However if {−1, −1} = 0 in K_{2}(F) then β^{2} is defined and c_{20} is onto because c_{20}(β^{2}) = ζ^{⊗2} is the generator of μ_{2}^{⊗2}.

Remark 5.2.2. let A be any commutative ring containing \frac{1}{q} and ζ. Then the assertion in 5.2 about c_{i0} remains valid for A. If P ic(A) has no q-torsion, the assertion in 5.2 about c_{i1} remains valid.

For the next result, recall from 2.6.3 that the restriction of c_{22} to SK_{2}(A; ℤ/q) is a homomorphism.

Proposition 5.3. Let A be a Dedekind domain containing \frac{1}{q} and a primitive q^{th} root of unity ζ_{q}.

(1) Assume that for every residue field k_{v} the map K_{2}(A) → K_{2}(k_{v}) ⊗ ℤ/q is onto. Then there is an exact sequence:

⊕K_{2}(k_{v}) ⊗ ℤ/q → SK_{2}(A; ℤ/q) ⟶^{c_{22}} H_{et}^{2}(A; μ_{q}^{⊗2}) → 0.

(2) If the residue fields of A are finite, there is an isomorphism:

SK_{2}(A; ℤ/q)⟶_{c_{22}}^{≅}H_{et}^{2}(A; μ_{q}^{⊗2}).

(3) If A lies in a global field then c_{22} : K_{2}(A) ⊗ ℤ/q ≅ H_{et}^{2}(A; μ_{q}^{⊗2}).

Proof. The result will follow from a diagram chase, once we establish that the diagram (5.3.1) is commutative with exact rows. Note that the maps c_{10} and c_{21} in (5.3.1) are onto, and the kernel of c_{10} is ⊕K_{2}(k_{v}) ⊗ ℤ/q; if the k_{v} are finite then c_{10} is an isomorphism. (5.3.1)

K_{3}(F; ℤ/q) ⟶^{∂} ⊕K_{2}(k_{v}; ℤ/q) ⟶^{τ} SK_{2}(A; ℤ/q) ⟶SK_{2}(F; ℤ/q) ⟶^{∂} ⊕k_{v}^{∗} ⊗ ℤ/q
↓^{c_{21}}  ↓^{−c_{10}}  ↓^{c_{22}}  ≅↓^{c_{22}}  ≅↓^{−c_{11}}
H^{1}(F; μ_{q}^{⊗2}) ⟶ ⊕H^{0}(k_{v}; μ_{q}) ⟶^{τ} H^{2}(A; μ_{q}^{⊗2}) ⟶ H^{2}(F; μ_{q}^{⊗2}) ⟶^{∂} ⊕H^{1}(k_{v}; μ_{q})

The rows are the (exact) localization sequences (4.0). The outer squares commute by 4.2(b), and the third square by naturality (2.1.1). To see that the second square commutes, we mimic the argument of 4.3. Fix β ∈ K_{2}(A; ℤ/q), and write β_{v} for its image in K_{2}(k_{v}; ℤ/q). Now [k_{v}] = 1 − [𝔭] in K_{0}(A), so if we set λ = c_{12}([k_{v}]) ∈ H_{et}^{2}(A; μ_{q}) the product formula 3.11(iv) yields

c_{22}τ(β_{v}) = c_{22}({[k_{v}], β}) = −λ ∪ c_{10}(β) = −τc_{10}(β_{v}).

Porism 5.3.2. (Suslin [Sus2, 4.4]). Writing H^{0}(A, K_{2}) for the image of K_{2}(A) in K_{2}(F), we may use 2.3.6 to rewrite 5.3 as the exact sequence

0 → H^{0}(A, K_{2}) ⊗ ℤ/q ⟶^{c_{22}} H_{et}^{2}(A; μ_{q}^{⊗2}) ⟶^{∂} _{q}SK_{1}(A) → 0.

Corollary 5.3.3. (Soulé [S, IV.1.5]). Let A be a Dedekind domain satisfying the hypotheses of 5.3. Then the cokernel of

c_{i2} : K_{2i−2}(A; ℤ/q) → H_{et}^{2}(A; μ_{q}^{⊗i}) ≅ H_{et}^{2}(A; μ_{q}^{⊗2}) ⊗ μ_{q}^{⊗(i−2)}

has exponent dividing (i − 1)!.

Proof. If q = 2 there is nothing to prove, so assume q ≢ 2 (mod 4). Let β ∈ K_{2}(A; ℤ/q) be a mod q Bott element in the sense of 2.7.2. If α ∈ K_{2}(A; ℤ/q) and i ≥ 3 we can cite 3.2(iv′) and 2.7.2 to see that

c_{i2}({α, β^{i−2}}) = (−1)^{i}(i − 1)!c_{22}(α) ⊗ [ζ]^{⊗i−2}.

Since c_{22} maps onto H_{et}^{2}(A; μ_{q}^{⊗i}) by 5.3, the result follows.

Corollary 5.4. (Soulé [S, IV.1.4]). Let A be a Dedekind domain whose field of fractions is a local or global field. Then for all q with \frac{1}{q} ∈ A:

c_{22} : K_{2}(A) ⊗ ℤ/q ⟶^{≅} H_{et}^{2}(A; μ_{q}^{⊗2}).

Proof. Since SK_{1}(A) = 0, this is 5.2 when μ_{q} ⊆ A. In particular, this follows when q is prime (using a transfer argument for odd primes). The general case follows from induction, as in [MS2, 11.5], using the (slightly modified) diagram of 2.8

K_{3}(A; ℤ/ℓ) ⟶K_{2}(A) ⊗ ℤ/ℓ^{ν} ⟶K_{2}(A) ⊗ ℤ/ℓ^{ν+1} ⟶K_{2}(A) ⊗ ℤ/ℓ ⟶0
↓^{c_{21}}  ↓^{c_{22}}  ↓^{c_{22}}  ↓^{c_{22}}
H^{1}(A; μ_{ℓ}^{⊗2}) ⟶ H^{2}(A; μ_{ℓ^{ν}}^{⊗2}) ⟶ H^{2}(A; μ_{ℓ^{ν+1}}^{⊗2}) ⟶ H^{2}(A; μ_{ℓ}^{⊗2})

and 5.5 below, which says that c_{21} is onto.

Corollary 5.4.1. (Soulé [S, IV.1.7]). Let F be a global field. Then the cokernel of c_{i2} : K_{2i−2}(A; ℤ/q) → H_{et}^{2}(A; μ_{q}^{⊗i}) has exponent dividing:

i! if F is totally imaginary, or
2i! if F has a real embedding.

Proof. If q is odd, we cite [S, IV.1.7], so we can assume that q = 2^{ν}. Let N : K_{∗}(F (ζ); ℤ/q) → K_{∗}(F; ℤ/q) be the transfer map. If x ∈ K_{2i−2}(F (ζ); ℤ/q) we know by 4.3 that

i{c_{i2}(Nx) − Nc_{i2}(x)} = 0.

If F is totally imaginary then the transfer map N : H_{et}^{2}(F (ζ); μ_{q}^{⊗i}) → H_{et}^{2}(F; μ_{q}^{⊗i}) is onto by [S, III.1.5]. By 5.3.3 the cokernel of c_{i2} has exponent dividing i!.

If F has a real embedding, the transfer factors as the surjection H_{et}^{2}(F (ζ); μ_{q}^{⊗i}) → H_{et}^{2}(F (\sqrt{−1}); μ_{q}^{⊗i}) followed by a degree 2 transfer. Hence the image contains 2H_{et}^{2}(F; μ_{q}^{⊗i}), and need not be onto. Therefore we lose another factor of 2.

Now suppose that A is a Dedekind domain with finite residue fields. Write K_{3}(A)_{ind} and K_{3}(A; ℤ/q)_{ind} respectively for the quotients of K_{3}(A) and K_{3}(A; ℤ/q) by the image of K_{3}^{M}(F) ⊕ ∐_{v}K_{3}(k_{v}). These are respectively subgroups of K_{3}(F)_{ind}

and K_{3}(F; ℤ/q)_{ind}, and there is a short exact sequence, split if q ≠ 2:

0 → K_{3}(A)_{ind} ⊗ ℤ/q → K_{3}(A; ℤ/q)_{ind} → T or(K_{2}(A), ℤ/q) → 0,

Theorem 5.5. Let A be a Dedekind domain with finite residue fields. Then c_{21} induces an isomorphism for all q:

K_{3}(A; ℤ/q)_{ind} ⟶^{≅} H_{et}^{1}(A; μ_{q}^{⊗2}).

Proof. The localization sequences (4.0) fit into the diagram

K_{3}^{M}(F)
↓
K_{3}(A; ℤ/q) ⟶ K_{3}(F; ℤ/q) ⟶ ⊕K_{2}(k_{v}; ℤ/q)
↓^{c_{21}}  ↓^{c_{21}}  ≅↓^{−c_{10}}
0 ⟶ H^{1}(A; μ_{q}^{⊗2}) ⟶ H^{1}(F; μ_{q}^{⊗2}) ⟶ ⊕H^{0}(k_{v}; μ_{q}^{⊗2})

which commutes by 2.2.1 and 4.2. The result for A follows from the corresponding isomorphism K_{3}(F; ℤ/q)_{ind} ≅ H^{1}(F, μ_{q}^{⊗2}) mentioned above.

Remark 5.5.1. If the residue fields are infinite, the appropriate statement involves K-cohomology, and is analogous to the analysis of 5.3.2. For example, the proof yields a natural surjection from H^{1}(A; μ_{q}^{⊗2}) onto the q-torsion subgroup of H^{0}(A, K_{2}), and the analysis of [Sus2, §5] applies.

Corollary 5.6. Let A be a Dedekind domain with finite residue fields. If A contains \frac{1}{q} and a primitive q^{th} root of unity, then the cokernel of

c_{i1} : K_{2i−1}(A; ℤ/q) → H_{et}^{1}(A; μ_{q}^{⊗i})

has exponent dividing (i − 1)!.

Proof. Combine the proofs of 5.2 and 5.3.

6. The e-invariant and Harris-Segal summands.

Let F be a field of characteristic 0, and F̄ its algebraic closure. By [Sus1], the torsion subgroup of K_{2i−1}(F̄) is ℚ/ℤ(i), the union over q of the Galois groups μ_{q}^{⊗i}. The map K_{2i−1}(F) → K_{2i−1}(F̄) must send the torsion subgroup _{tor}K_{2i−1}(F) of K_{2i−1}(F) to H_{et}^{0}(F; ℚ/ℤ(i)). We shall use the term e-invariant to describe either the map (which is well-defined up to automorphism of ℚ/ℤ(i))

torK_{2i−1}(F) → H_{et}^{0}(F; ℚ/ℤ(i))

or the map it induces on mod q K-theory, i.e., the horizontal maps of

K_{2i}(F; ℤ/q) ⟶^{e} H_{et}^{0}(F; μ_{q}^{⊗i}) ⊆ μ_{q}^{⊗i}(F̄)= K_{2i}(F̄; ℤ/q)
↓  ↓  ↓
torK_{2i−1}(F) ⟶ H_{et}^{0}(F; ℚ/ℤ(i))⊆ ℚ/ℤ(i).

By [Q], the map e is a generalization of Adam’s (complex) e-invariant, whence the name. The e-invariant for i = 2 is the same as the map “d” of [MS, §9].

If F is a field of characteristic p ≠ 0, with separable closure F̄, we can repeat the above by restricting to those q not divisible by p. In this case the e-invariant refers either to the map e : K_{2i}(F; ℤ/q) → H_{et}^{0}(F; μ_{q}^{⊗i}) ⊆ K_{2i}(F̄; ℤ/q) or to the map torK_{2i−1}(F) → ℚ/ℤ[\frac{1}{p}](i). Our interest in the e-invariant stems from the following result.

Proposition 6.1. If F is a field containing \frac{1}{q}, then c_{i0} : K_{2i}(F; ℤ/q) → H_{et}^{0}(F; μ_{q}^{⊗i}) is (−1)^{i−1}(i − 1)! times the e-invariant. In particular, the integral Chern class c_{i0} : K_{2i}(F) → H_{et}^{0}(F; μ_{q}^{⊗i}) is always zero.

Proof. If F = F̄ this follows from 1.6.2 and the product formula, as in the proof of 5.1. The result for general F follows by naturality:

K_{2i}(F; ℤ/q) →  K_{2i}(F̄; ℤ/q)
↓ c_{i0}  ↓ c_{i0}
H_{et}^{0}(F; μ_{q}^{⊗i}) ↪ μ_{q}^{⊗i} ≅ ℤ/q.

Remark 6.1.1. For any scheme X over ℤ[\frac{1}{q}], μ_{q}^{⊗i} is isomorphic to the étale sheaf associated to the presheaf U ↦ K_{2i}(U; ℤ/q). (This observation follows from Theorem 1 of [Gab].) Therefore we may define the e-invariant on X to be the resulting map

K_{2i}(X; ℤ/q) → _{q}K_{2i−1}(X) → H_{et}^{0}(X; μ_{q}^{⊗i}).

Corollary 6.1.2. If A is any commutative ring containing \frac{1}{q} then the integral Chern class c_{i0} : K_{2i}(A) → H_{et}^{0}(A; μ_{q}^{⊗i}) is zero, and Soulé’s Chern class factors through the q-torsion subgroup _{q}K_{2i−1}(A) of K_{2i−1}(A). In fact, c_{i0} is (−1)^{i−1}(i−1)! times the e-invariant.

Example 6.2. Let F be a global field, and consider the induced map c_{20} from qK_{3}(F) to H^{0}(F; μ_{q}^{⊗2}) ≅ ℤ/(q, w_{2}). Replacing F by F (ζ) and using 5.2, we see that c_{20} is the e-invariant; see [MS,§9]. By Theorem 10.2 of [MS] we have:

• If F is totally imaginary or if q is odd we have an isomorphism:

c_{20} : _{q}K_{3}(F) ⟶^{≅} H_{et}^{0}(F; μ_{q}^{⊗2})

• If F has r_{1} ≠ 0 real embeddings and q = 2^{ν}, the kernel of c_{20} is the Milnor K-group K_{3}^{M}(F) ≅ (ℤ/2)^{r_{1}}. If ν is sufficiently large, then H_{et}^{0}(F; μ_{q}^{⊗2}) ≅ ℤ/w_{2} is independent of q and K_{3}(F) ≅ (ℤ/2w_{2}) ⊕ (ℤ/2)^{r_{1}−1} ⊕ ℤ^{r_{2}}. Hence c_{20} is onto for q ≥ 2w_{2} but if q ≤ w_{2} there is a non-split exact sequence

0 → K_{3}^{M}(F) → _{q}K_{3}(F) ⟶^{c_{20}} H_{et}^{0}(F; μ_{q}^{⊗i}) → ℤ/2 → 0.

It is useful to have a formula for the order w_{i} = w_{i}(F) of H_{et}^{0}(F; ℚ/ℤ(i)), or rather for the order w_{i}^{(ℓ)} of its ℓ-Sylow subgroup.

First of all, suppose that ℓ ≠ 2 and let 0 ≤ m ≤ ∞ be maximal such that F contains a primitive ℓ^{th}-root of unity ζ_{ℓ^{m}}. The following formula is due to Harris and Segal [HS]. If ζ_{ℓ} ∈ F then ω_{i}^{(ℓ)} = ℓ^{m+λ}, where i = ℓ^{λ}s with s prime to ℓ. If ζ_{ℓ} ∉ F then

w_{i}^{(ℓ)}(F) = \{ \begin{matrix}w_{i}^{(ℓ)}(F (ζ_{ℓ})) & if i ≡ 0 mod (ℓ − 1) \\ 1 & otherwise. \end{matrix}

Definition 6.3.0. For the order w_{i}^{(2)}of the 2-Sylow subgroup, we need to introduce some notation due to Harris and Segal [HS]. Call a field F exceptional if the Galois group Gal(F (ζ)/F) is not cyclic for some primitive 2^{ν} root of unity ζ, and nonexceptional otherwise. If F has a real embedding then F is exceptional, while if \sqrt{−1} ∈ F then F is non-exceptional. A field F is non-exceptional iff there is a 2-primary root of unity ζ ≠ ±1 such that ζ − ζ^{−1} ∈ F (see [K]).

Lemma 6.3. (cf. [HS,p.28]) let F be a field of characteristic ≠ 2.

(a) If F contains ζ_{4} = \sqrt{−1}, write i = 2^{λ}s with s odd. If m is maximal such that F contains a primitive 2^{m}-root of unity then w_{i}^{(2)}(F) = 2^{m+λ}.

(b) If \sqrt{−1} ∉ F and i is odd then w_{i}^{(2)}(F) = 2.

(c) If \sqrt{−1} ∉ F, F is exceptional and i is even then w_{i}^{(2)}(F) = w_{i}^{(2)}(F (\sqrt{−1}))

(d) If \sqrt{−1} ∉ F, F is non-exceptional and i is even then

w_{i}^{(2)}(F) = \frac{1}{2}w_{i}^{(2)}(F (\sqrt{−1})).

Corollary 6.3.1. When F = ℚ and i = 2^{λ}s with s odd, then

w_{i}^{(2)}(ℚ) = \{ \begin{matrix}8 · 2^{λ−1} & if i is even \\ 2 & if i is odd \end{matrix}

If A is a Dedekind domain with field of fractions F, we have H_{et}^{0}(A; μ_{q}^{⊗i}) ≅ H_{et}^{0}(F; μ_{q}^{⊗i}). If F is a local or global field then by 4.6 and 6.1.2 the map c_{i0} factors through _{q}K_{2i−1}(A) = _{q}K_{2i−1}(F). When i = 2 and r_{1} ≠ 0, 6.2 shows that the cokernel of c_{i0} depends on q.

Theorem 6.4. (Harris-Segal). If F is a non-exceptional global field, the e-invariant is naturally split, so that there are natural cyclic direct summands E^{i} of K_{2i−1}(F) isomorphic to ℤ/w_{i}(F). In particular, this is the case if char(F) ≠ 0 or if F is a number field containing \sqrt{−1}.

Proof. Let S be a finite nonempty set of places of F and A the ring of S-integers in F. By 4.6 the group K_{2i−1}(A) ≅ K_{2i−1}(F) is independent of S. Harris and Segal proved in [HS] that K_{2i−1}(A) has a direct summand E^{i}(F) isomorphic to ℤ/w_{i}(F). If F ⊂ F′ is a finite separable extension, their proof shows that E^{i}(F) injects into E^{i}(F′). Now take the direct limit over all F′ to see that E^{i}(F) injects into the torsion subgroup of K_{2i−1}(F̄); this map is the e-invariant.

Now suppose that F is exceptional, hence a number field. Let E_{(ℓ)}^{i} denote the ℓ-Sylow subgroup of E^{i}(F (\sqrt{−1})), and τ_{∗} : K_{2i−1}(F (\sqrt{−1})) → K_{2i−1}(F) the transfer map. If ℓ ≠ 2, E_{(ℓ)}^{i} maps isomorphically onto a summand E_{(ℓ)}^{i}(F) of K_{2i−1}(F); if ℓ = 2 all we can say is that τ^{∗} : K_{2i−1}(F) → K_{2i−1}(F (\sqrt{−1})) maps τ_{∗}(E_{(2)}^{i}) onto the subgroup 2E_{(2)}^{i} [HS,p.30]. From this, Harris and Segal deduce the following result.

Corollary 6.4.1. (Harris-Segal) If F is an exceptional number field then there is a natural cyclic direct summand E^{i} of K_{2i−1}(F) of order either w_{i}, 2w_{i} or w_{i}/2.

In the rest of this section we make some improvements upon the uncertainty in the 2-Sylow subgroup of E^{i}.

First suppose that i is odd, so that w_{i}^{(2)}(F) = 2; Harris and Segal’s assertion about E_{(2)}^{i} is vacuous in this case as τ^{∗}τ_{∗}E_{(2)}^{i}(F (\sqrt{−1})) = 0.

If i = 4k + 1, so that 2i − 1 = 8k + 1, we define E_{(2)}^{i}(F) to be the subgroup of K_{8k+1}(F) generated by the image of the Adams element η_{8k+1} of order 2 in π_{8k+1}^{s}. Browder proved in [Br,4.7] (cf. [Q]) that this element maps nontrivially into K_{8k+1}(ℂ), so E_{(2)}^{i}(F) ≅ ℤ/2. As the target of the e-invariant is the subgroup ℤ/2 of K_{8k+1}(ℚ̄), E_{(2)}^{i}(F) is a direct summand of K_{8k+1}(F). In this case we define the Harris-Segal summand to be E^{i} = E_{odd}^{i} ⊕ E_{(2)}^{i}; it is mapped by the e-invariant isomorphically onto H_{et}^{0}(F; ℚ/ℤ(i)) ≅ ℤ/w_{i}.

If i = 4k+3, so that 2i−1 = 8k+5, we consider two cases. If F has a real embedding, then w_{i}(F) = 2. The e-invariant K_{8k+5}(F) → ℤ/2 is zero because it factors through the group K_{8k+5}(ℝ), which is uniquely divisible by [Sus1]. Since the image of K_{8k+5}(F) in K_{8k+5}(F (\sqrt{−1})) misses E^{i}(F (\sqrt{−1})) and τ^{∗}τ_{∗} kills E^{i}(F (\sqrt{−1})), we may as well define E^{i}(F) to be zero.

If F is totally imaginary the situation is completely different, as the following result shows: the e-invariant is an isomorphism, provided we define the 2-Sylow subgroup of E^{i} properly.

Proposition 6.5. If F is a totally imaginary number field, then for every odd i there is a natural cyclic direct summand E^{i} of K_{2i−1}(F) of order w_{i}(F), mapped isomorphically by the e-invariant to H_{et}^{0}(F; ℚ/ℤ(i)) ≅ ℤ/w_{i}. The 2-Sylow subgroup of E^{i} has order 2.

Proof. Given 6.4.1 and 6.3(b), it suffices to find a naturally defined element of order 2 in K_{4k+2}(F; ℤ/8) which is detected by the e-invariant. Let β ∈ K_{2}(ℂ; ℤ/8) be a mod 8 Bott element, and γ ∈ K_{2}(ℤ; ℤ/8) ≅ ℤ/2 ⊕ ℤ/2 an element mapping to 4β in K_{2}(ℂ; ℤ/8). Let z ∈ K_{3}(ℤ) be an element of order 16 mapping to ∂(β^{2}) in K_{3}(ℂ). By [MS] the image z̄ of z in K_{3}(F) has order 8, so there also exists an element ω ∈ K_{4}(F; ℤ/8) with ∂ω = z̄. By construction, the image of ω in K_{4}(ℂ; ℤ/8) is β^{2}. The product ω^{k}γ ∈ K_{4k+2}(F; ℤ/8) maps to the nonzero element 4β^{2k+1} of K_{4k+2}(ℂ; ℤ/8) ≅ ℤ/8, so ω^{k}γ is detected by the e-invariant. Since 2(ω^{k}γ) = ω^{k}(2γ) = 0, ω^{k}γ is our sought-for element.

We now turn to the situation when i = 2n is even. We first describe the situation when F is totally imaginary.

Theorem 6.6. Let F be a totally imaginary number field and w_{i} = w_{i}(F). Then

(a) If i = 4k + 2, the Harris-Segal summand E^{i} of K_{8k+3}(F) has order w_{i}, and the e-invariant induces an isomorphism

e : E^{i} ≅ H_{et}^{0}(F; ℚ/ℤ(i)) ≅ ℤ/w_{i}.

(b) If i = 4k, the Harris-Segal summand E^{i} of K_{8k−1}(F) has order either w_{i} or \frac{1}{2}w_{i}, and the e-invariant maps E^{i} injectively into H_{et}^{0}(F; ℚ/ℤ(i)) ≅ ℤ/w_{i}.

Proof. Given 6.4.1, it suffices to determine the 2-Sylow subgroup of E^{i}. Set q = w_{2}^{(2)}(F), and let β ∈ K_{2}(ℂ; ℤ/q) be a mod q Bott element. By 6.2 there is an element ω ∈ K_{4}(F; ℤ/q) mapping to β^{2} ∈ K_{4}(ℂ; ℤ/q). The element ω^{n} of K_{4n}(F; ℤ/q) maps to β^{2n}, so both ω^{n} and ∂(ω^{n}) ∈ K_{4n−1}(F; ℤ/q) have order q and are mapped by the e-invariant to an element of order q. This proves (b). If n is odd then w_{2n}^{(2)}(F) = q, and ∂(ω^{n}) generates a cyclic summand of order q in K_{4n−1}(F). This proves (a).

Remark 6.6.1. If F is totally imaginary then by [DF, 5.2] the 2-primary étale Ktheory of Spec(F) fits into an extension

0 → H_{et}^{2}(F; μ_{q}^{⊗i+1}) → K_{2i}^{et}(F, ℤ/q) → H_{et}^{0}(F; μ_{q}^{⊗i}) → 0.

There is a natural map ρ : K_{2i}(F; ℤ/q) → K_{2i}^{et}(F; ℤ/q), and it is not hard to see that the e-invariant is the composition of ρ with the above surjection. If \sqrt{−1} ∈ F, then ρ is onto by [DF, 8.2]. In general, the surjectivity of ρ would imply that the Harris-Segal summand E^{4k} of K_{8k−1}(F) has order w_{4k}.

Remark 6.6.2. It seems reasonable that for a totally imaginary number field F the Harris-Segal summand E^{i} of K_{2i−1}(F) is mapped isomorphically by the e-invariant onto H_{et}^{0}(F; ℚ/ℤ(i)) ≅ ℤ/w_{i}. By 6.4, 6.5 and 6.6 this is true unless 4|i. The next result implicitly establishes this result for any F such that F (\sqrt{−1}) contains no primitive 8^{th} root of unity.

Theorem 6.7 (F = ℚ). For every n ≥ 1, let w = w_{2n}(ℚ) denote the order of the cyclic group H_{et}^{0}(ℚ; ℚ/ℤ(2n)). Then:

(a) Im(J)_{4n−1} ≅ ℤ/w.

(b) The denominator of ζ(1 − 2n) is \frac{1}{2}w.

(c) The Harris-Segal direct summand E^{2n} of K_{4n−1}(ℤ) = K_{4n−1}(ℚ) contains the image of J.

(d) If 4n − 1 ≡ 7(mod 8) then E^{2n} = Im(J)_{4n−1} ≅ ℤ/w and the e-invariant induces an isomorphism

e : E^{2n} ≅ H_{et}^{0}(ℚ; ℚ/ℤ(2n)).

(e) If 4n − 1 ≡ 3(mod 8) then E^{2n} is cyclic of order 2w and the e-invariant induces a short exact sequence:

0 → ℤ/2 → E^{2n} ⟶^{e} H_{et}^{0}(ℚ; ℚ/ℤ(2n)) → 0.

Proof. Let B_{n} denote the n^{th} Bernoulli number. It is well-known that ζ(1 − 2n) = −B_{n}/2n, and that Im(J)_{4n−1} is a cyclic group of order equal to the denominator of B_{n}/4n; see [CC,p.284]. Since the numerator of B_{n} is odd, this shows that (a) implies (b). The calculation of the denominator of B_{n}/n on p. 284 of [CC], compared with the calculation of w_{2n} in 6.3.1, proves (a). By [Q], Im(J)_{4n−1} injects into K_{4n−1}(ℤ), and is a direct summand if 4n − 1 ≡ 7(mod 8). The construction used by Harris and Segal shows that Im(J)_{4n−1} is contained in E^{2n} (see 3.2 and p. 30 of [HS], recalling that 16|w_{2n}). Hence Im(J)_{4n−1} = E^{2n}, except possibly for the 2-Sylow subgroup when 4n − 1 ≡ 3(mod 8). The 2-Sylow subgroup of Im(J)_{8k+3} is ℤ/8, and is contained in a ℤ/16 summand of K_{8k+3}(ℤ) by [Br,4.8]. Hence the 2-Sylow subgroup of E^{2n} must be that ℤ/16 summand.

Application 6.7.1. If q = 2^{ν} the 2-primary Chern class

c_{2n,0} : K_{4n}(ℚ; ℤ/q) → _{q}E^{2n} → H_{et}^{0}(ℚ; μ_{q}^{⊗2n}) ≅ ℤ/(w_{2n}, q)

is zero unles n is 1, 2 or 4.

• (n = 1) E^{2} = K_{3}(ℚ) ≅ ℤ/48, and the kernel of

c_{20} : ℤ/(48, q) = _{q}(E^{2}) → ℤ/(24, q)

has order 2. This map is onto iff 16|q.

• (n = 2) E^{4} ⊆ K_{7}(ℤ) is isomorphic to ℤ/w_{4} = ℤ/240 ≅ ℤ/16 ⊕ ℤ/15 and c_{40} may be identified with multiplication by 6.

• (n = 4) E^{8} ⊆ K_{15}(ℤ) is isomorphic to ℤ/w_{8} = ℤ/480 ≅ ℤ/32 ⊕ ℤ/15 and c_{80} may be identified with multiplication by 16 · 15. This map is nonzero iff 32|q.

Remark 6.7.2. Browder asserted in [Br,4.8] that the ℤ/48 in K_{3}(ℤ) reproduces itself periodically as a direct summand of every K_{8k+3}(ℤ). However, the ℤ/3 is only a summand if k ≢ 1 mod 3. For example, the summand of K_{11}(ℤ) is

E^{6} ≅ ℤ/16 ⊕ ℤ/9 ⊕ ℤ/7.

Corollary 6.8. Let F be a number field with a real embedding, and w_{i} = w_{i}(F).

(a) If i = 4k + 2, the Harris-Segal summand E^{i} of K_{8k+3}(F) is cyclic of order w_{i} or 2w_{i}, and injects into K_{8k+3}(ℝ). Hence the kernel of e : E^{i} → H_{et}^{0}(F; ℚ/ℤ(i)) has order 2.

(b) If i = 4k, the Harris-Segal summand E^{i} of K_{8k−1}(F) is cyclic of order w_{i} or \frac{1}{2}w_{i}, and the e-invariant maps E^{i} injectively into H_{et}^{0}(F; ℚ/ℤ(i)) ≅ ℤ/w_{i}.

Proof. The composition E^{i}(ℚ) → E^{i}(F) → E^{i}(ℝ) = ℚ/ℤ is an injection by 6.7, and E^{i}(ℚ) is cyclic of even order. Therefore, the 2-Sylow subgroup of the cyclic group E^{i}(F) must inject into E^{i}(ℝ). As the odd subgroup of E^{i}(F) injects by Harris and Segal’s result 6.4, E^{i}(F) must inject into E^{i}(ℝ). Now E^{i}(ℝ → E^{i}(ℂ) is an isomorphism if i = 4k, but a surjection with kernel ℤ/2 if i = 4k + 2. Since the image of E^{i}(F) in E^{i}(ℂ) has order w_{i} or \frac{1}{2}w_{i}, the result follows.

Remark 6.8.1. If F has a real embedding, we have shown that the “real e-invariant” e_{ℝ} : E^{i}(F) → E^{i}(ℝ) ≅ ℚ/ℤ(i) is an injection for all i. It is not hard to see that this map is independent of the choice of real embedding, even though this choice affects the extension of e_{ℝ} to the entire torsion subgroup

torK_{2i−1}(F) → E^{i}(ℝ) ≅ ℚ/ℤ(i).

It seems reasonable to expect that the order of E^{i}(F) is: 2w_{i} if i = 4k + 2, and w_{i} if i = 4k. That is, we should expect the real e-invariant to be an isomorphism:

e_{ℝ} : E^{i}(F) ≅ H^{0}(F; E^{i}(ℝ)).

7. Towards the cohomology conjectures.

Let F be a number field of characteristic ≠ 2 and write A for O_{F} [\frac{1}{2}]. Lichtenbaum’s conjectures at the prime ℓ = 2 involve the 2-Sylow subgroup W^{(i)} of ℚ/ℤ(i). Note that W^{(i)} is also the étale sheaf lim_{ν→∞}μ_{2^{ν}}^{⊗i} of all 2-primary roots of unity, Tate twisted i times. As noted in III.1.3 of [S], the inclusion j : Spec(F) → Spec(A) induces isomorphisms j_{∗}μ_{q}^{⊗i} ≅ μ_{q}^{⊗i} and hence j_{∗}W^{(i)} ≅ W^{(i)}.

Lichtenbaum conjectured in [Licht] that : (L1) if F is totally real then H_{et}^{k}(A; W^{(i)}) = 0 for k ≥ 2 and i ≥ 2 even, and (L2) the 2-part of ζ_{F} (−1) equals the Euler characteristic

χ = ^{#}H_{et}^{1}(A; W^{(i)})/^{#}H_{et}^{0}(A; W^{(i)}).

Lichtenbaum’s analogous conjectures for odd primes ℓ have been verified. For odd ℓ, cd_{ℓ}(A) = 2 implies (L1) except for the vanishing of H^{2} which was proven by Soulé in [S,IV.3.2]. Recently, A. Wiles proved the Main Conjecture of Iwasawa Theory in [Wiles], which implies that the ℓ-part of ζ_{F} (−1) equals the corresponding ℓ-primary Euler characteristic.

The case i = 2 of (L2) is equivalent to (the 2-primary part of) the Birch-Tate conjecture that ^{#}K_{2}(A) = w_{2}(F)|ζ_{F} (−1)|. Indeed, H^{0}(A; W^{(2)}) ≅ (ℤ/w_{2}(F))_{(2)} and K_{2}(A)_{(2)} ≅ H^{1}(A; W^{(2)}) by [Sus2,3.11]. The Birch-Tate conjecture has been verified for many families of totally real F by Hurrelbrink and Kolster.

It is an easy calculation using [MS2] that H^{2}(A, W^{(2)}) = 0, but the following result shows that H^{3}(A; W^{(2)}) ≅ (ℤ/2)^{r_{1}}. Hence (L1) fails.

Proposition 7.1. Let F be a number field with r_{1} real embeddings. Then for k ≥ 3 and i ≥ 1:

H_{et}^{k}(A; W^{(i)}) ≅ H_{et}^{k}(F; W^{(i)}) ≅ \{ \begin{matrix}(ℤ/2)^{r_{1}} & if k + i is odd \\ 0 & if k + i is even \end{matrix}

Proof. If k is a finite field then H_{et}^{k}(k; W^{(i)}) = 0 for k ≥ 2. If k = 1 and i ≥ 1 then H^{1}(k; W^{(i)}) = 0 by 5.1. The group H^{1}(k; W^{(0)}) is W^{(0)} ≅ ℚ_{2}/ℤ_{2}. Since \frac{1}{2} ∈ A, the map from Br(F) ≅ H^{2}(F; W^{(1)}) to ⊕H^{1}(k_{v}; W^{(0)}) is onto by local class field theory. Therefore the localization sequence (4.0) yields H^{k}(A; W^{(i)}) ≅ H^{k}(F; W^{(i)}) for k ≥ 3. On the other hand, Tate proved (see [AV] [Tate]) that the natural map

H_{et}^{k}(F; W^{(i)}) → ⊕_{r_{1}}H_{et}^{k}(ℝ; W^{(i)})

is an isomorphism for k ≥ 3 and a surjection for k = 2. Therefore it suffices to compute H_{et}^{∗}(ℝ; W^{(i)}), which we do in the following lemma.

Lemma 7.1.1.

H_{et}^{k}(ℝ; W^{(i)}) ≅ \{ \begin{matrix}W^{(i)} & if k = 0 and i is even \\ ℤ/2 & if k ≥ 0 and k + i is odd \\ 0 & if k > 0 and k + i is even \end{matrix}

Proof. Let σ denote complex conjugation. If i is even then σ = 1 on W^{(i)}; if i is odd then σ = −1 on W^{(i)}. The Lemma follows from the observation that H^{∗}(ℝ, W^{(i)}) is the cohomology of the periodic chain complex:

0 ⟶ W^{(i)} ⟶^{1−σ} W^{(i)} ⟶^{1+σ} W^{(i)} ⟶^{1−σ} · · · .

Remark 7.1.2. In case i = 1 we have an isomorphism

H^{k}(A; W^{(1)}) ≅ 2-torsion in H^{k}(A; 𝔾_{m}) ⊕ H^{k−1}(A; 𝔾_{m}) ⊗ W^{(0)}
≅ H^{k}(A; 𝔾_{m})_{(2)} for k ≠ 1.

Thus if A has n primes over 2 we have

H^{2}(A; W^{(1)}) ≅ Br(A)_{(2)} ≅ (ℤ/2)^{r_{1}} ⊕ (ℚ/ℤ)^{n−1},

and H^{3}(A; 𝔾_{m}) = 0, H^{4}(A; 𝔾_{m}) ≅ (ℤ/2)^{r_{1}}, etc. The reader should compare this result with the discussion on pp.108-9 of [Milne].

Remark 7.1.3. Recently, Levine [L] and Merkurjev-Suslin [MS] proved that for a totally real number field F we have a (non-split) extension

0 ⟶ (ℤ/2)^{r_{1}} ⟶ K_{3}(F) ⟶^{c_{21}} H^{0}(F; ℚ/ℤ(2)) ⟶ 0.

Since K_{3}(A) = K_{3}(F) by 4.6 and c_{21} : K_{2}(A) ≅ H_{et}^{1}(A; ℚ/ℤ(2)) by 5.5 we see that

\frac{^{#}K_{2}(A)}{^{#}K_{3}(A)} ≅ \frac{^{#}H^{1}(A; ℚ/ℤ(2))}{^{#}H^{0}(A; ℚ/ℤ(2)) ·^{#}H^{4}(A; ℚ/ℤ(1))}

If the Birch-Tate conjecture holds for F then this equals ±2^{−r_{1}}ζ_{F} (−1). As observed in [L], this is related to the γ-filtration on K-theory. It also suggests that perhaps the 2-part of^{#}K_{2i−2}(A)/^{#}K_{2i−1}(A) is related to the étale Euler characteristic more closely than has been thought.

In order to describe H^{2}(A; W^{(i)}), we need to be more careful. By 7.1.2, we need to assume that i ≥ 2 in order to get manageable results. The proof of 7.1 shows that if i is odd then there are natural surjections

H^{2}(A; W^{(i)}) → H^{2}(F; W^{(i)}) → (ℤ/2)^{r_{1}}.

In order to proceed further, we need to distinguish between the exceptional and non-exceptional number fields (see 6.3.0).

Lemma 7.2. (Lichtenbaum) If F is non-exceptional and i ≥ 2 then H_{et}^{2}(F; W^{(i)}) = 0.

Proof. ([Licht,9.5]). Let F_{∞} = lim_{ν→∞}F (μ_{2^{ν}}) be the maximal abelian 2-primary extension of F, and let Γ denote the Galois group of F_{∞}/F. If F is non-exceptional, then Γ is cyclic, so that cd_{2}(Γ) = 1. Now F_{∞}^{∗} is a discrete Γ-module, and i ≥ 2, so by [T342] we have

H^{1}(Γ, H_{et}^{1}(F_{∞}; W^{(i)})) = H^{1}(Γ, F_{∞}^{∗} ⊗ W^{(i−1)}) = 0.

The Hochschild-Serre spectral sequence therefore degenerates. Since cd_{2}(F_{∞}) = 1 (see [S-CG,Prop.II-9]), this yields

H^{2}(F; W^{(i)}) = H^{2}(F_{∞}; W^{(i)})^{Γ} = 0.

Remark 7.2.1. If F is exceptional and i ≥ 2, the usual transfer argument shows that H^{2}(F; W^{(i)}) has exponent 2, and 2-rank ≥ r_{1} if i is odd.

Theorem 7.3. (Soulé). If F is totally imaginary (or a global field of characteristic ≠ 0, 2) and i ≥ 2 then H_{et}^{2}(A; W^{(i)}) = H_{et}^{2}(F; W^{(i)}) = 0.

Proof. ([S,IV.3.2]). By 4.2, we have a commutative diagram:

lim_{ν→∞}K_{2i−1}(F; ℤ/2^{ν}) ⟶^{∂} lim_{ν→∞}⊕_{v}K_{2i−2}(k_{v}; ℤ/2^{ν})
↓^{c_{i1}}  ↓^{(1−i)c_{i−1,0}}
H^{1}(F; W^{(i)})  ⟶^{∂} ⊕H^{0}(k_{v}; W^{(i−1)}) ⟶H^{2}(A; W^{(i)}) → H^{2}(F; W^{(i)}) → 0.

Since K_{2i−2}(k_{v}; ℤ/2^{ν}) ≅ K_{2i−1}(k_{v}) for ν ≫ 0 and K_{2i−2}(F) is a torsion group, the top map ∂ is onto by 4.6. By 5.1, the image of (1 − i)c_{i−1,0} is (i − 1)! times ⊕H^{0}(k_{v}; W^{(i−1)}). By 7.2 and 7.2.1 this implies that H^{2}(A; W^{(i)}) has exponent dividing 2(i − 1)!. But H^{2}(A; W^{(i)}) is divisible by the following lemma, so it must be zero.

Lemma 7.3.1. Let F be a totally imaginary number field (or a function field in one variable over a finite field of characteristic ≠ 2), and A a ring of S-integers in F containing \frac{1}{2}. Then for all i the group H_{et}^{2}(A, W^{(i)}) is divisible.

Proof. By [Tate,p.292], cd_{2}(A) = 2. Thus H^{3}(A; μ_{2}) = 0. Now consider the Kummer sequence associated to W^{(i)}:

H^{2}(A; W^{(i)}) ⟶^{2} H^{2}(A; W^{(i)}) ⟶ H^{3}(A; μ_{2}).

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